a bibliography of gamma function and related … · matematika](6):3{8, 90, 1987. russian. [6] c....

A BIBLIOGRAPHY OF GAMMA FUNCTION AND RELATED TOPICS

Version 0.5. – October 2011, http://milanmerkle.com

The original entries in this bibliography were created by merging the bibliographyof Jozsef Sandor with mine, resulting in the Version 0.4 with total of 892 items.Every item entered after Version 0.4 is accompanied by a note that contains thename of a person who submitted it and the date of submission.

Milan Merkle

References

[1] Ibrahim A. Abou Tair. On certain Dirichlet series related to Hurwitz Zeta-function. J. Inst.

Math. Comput. Sci. Math. Ser. [Journal of Institute of Mathematics and Computer Sciences(Mathematics Series)] 3(3):299–304, 1990.

[2] Jorge Alberto Achcar, Heleno Bolfarine. The log-linear model with a generalized Gamma

distribution for the error: A Bayesian approach. Statist. Probab. Lett. [Statistics and Prob-ability Letters] 4(6):325–332, 1986.

[3] V. S. Adamchik, O. I. Marichev. Representations of functions of hypergeometric type in

logarithmic cases. Vestsi Akad. Navuk BSSR Ser. Fiz. Mat. Navuk [Vestsi Akademii NavukBSSR. Seryya Fizika Matematychnykh Navuk](5):29–35, 1983. In Russian.

[4] A. Adatia, A. G. Law, Q. Wang. Characterization of a mixture of Gamma distributions viaconditional finite moments. Comm. Statist. Theory Methods [Communications in Statistics.

Theory and Methods] 20(5–6):1937–1949, 1991.

[5] I. I. Adgamov, I. N. Volodin. On a test for the Weibull distribution against a family of gener-alized Gamma-alternatives. Izv. Vyssh. Uchebn. Zaved. Mat. [Izvestiya Vysshikh Uchebnykh

Zavedenii. Matematika](6):3–8, 90, 1987. Russian.

[6] C. Adiga, T. Kim. On a generalization of Sandor’s function. Proc. Jangjeon Math. Soc.5(2):121–124, 2002.

[7] C. Adiga et al. On a q-analogue of Sandor’s function. J. Ineq. Pure Appl. Math. 4(5), 2003.

art.84 (electronic).[8] Alan Adolphson. Uniqueness of γp in the Gross-Koblitz formula for Gauss sums. Trans.

Amer. Math. Soc. [Transactions of the American Mathematical Society] 278(1):57–63, 1983.

[9] A.U. Afuwape, C.O. Imoru. Bounds for the Beta function. Bolletino U.M.I. 5(17-A):330–334, 1980.

[10] R.P. Agarwal. Difference equations and inequalities. Marcel Dekker, Inc., 2nd edition, 2000.[11] Abdul Hadi Nabih Ahmed, A. M. Abouammoh. Characterizations of Gamma, inverse Gauss-

ian, and negative binomial distributions via their length-biased distributions. Statist. Papers

[Statistical Papers. Statistische Hefte] 34(2):167–173, 1993.[12] A. M. Al Rashed, S. I. Ahmed. A generalization of the number π. J. Natur. Sci. Math. [The

Journal of Natural Sciences and Mathematics] 29(1):29–37, 1989.

[13] M. Masoom Ali, A. K. Md. E. Saleh, Dale Umbach. Estimating functions of location andscale parameters. Soochow J. Math. [Soochow Journal of Mathematics] 19(3):259–270, 1993.

[14] A. Alikhani, M. Hassani. Approximation of pn by hn. RGMIA Research Report Collection8(4), 2005.

[15] Giampietro Allasia, Renata Besenghi. Numerical calculation of incomplete gamma functions

by the trapezoidal rule. Numer. Math. [Numerische Mathematik] 50(4):419–428, 1987.

[16] Giampietro Allasia, Renata Besenghi. Numerical calculation of the Gamma and Digammafunctions using the trapezoidal rule. Boll. Un. Mat. Ital. B (7) [Unione Matematica Italiana.

Bollettino. B. Serie-VII] 1(3):815–828, 1987. Italian.[17] J.P. Allouche. Transcendence of the Carlitz-Goss Gamma function at rational arguments.

J. Number Theory 60(2):318–328, 1996.

[18] C. Alsina, M.S. Tomas. A geometrical proof of a new inequality for the Gamma function.J. Ineq. Pure Appl. Math. 6(2), 2005.

1

2 Bibliography of Gamma function–v.0.5 (October 2011) http://milanmerkle.com

[19] A.A. Alyakrinskii. On the representation of the values of Euler’s Gamma function at somerational points in the form of an infinite product. In Investigations in complex analysis

(Russian), 123–128. Krasnoyarsk. Gos. Univ., 1989.

[20] H. Alzer. On some inequalities involving (n!)1/n, II. Period. Math. Hung. 28(3):229–233,1994.

[21] H. Alzer. Note on an inequality involving (n!)1/n. Acta Math. Univ. Comen. New Ser.

64:283–285, 1995.[22] H. Alzer. Characterizations of the ratio of Gamma and q-Gamma functions. Abh. Math.

Sem. Univ. Hamburg 70:165–174, 2000.

[23] H. Alzer. Inequalities for the Gamma function. Proc. Amer. Math. Soc. 128:141–147, 2000.[24] H. Alzer. A mean-value inequality for the Gamma function. Applied Math. Letters 13:111–

114, 2000.

[25] H. Alzer. A power mean inequality for the Gamma function. Monatsh. Math. 131:179–188,2000.

[26] H. Alzer. Sharp bounds for the ratio of q-Gamma functions. Math. Nachr. 222:5–14, 2001.[27] H. Alzer. Sharp inequalities for the Beta function. Indag Math. (N.S.) 12:15–21, 2001.

[28] H. Alzer. Inequalities for the Beta function of n variables. The ANZIAM Journal 44:609–

623, 2003.[29] H. Alzer. On Ramanujan’s double inequality for the Gamma function. Bull. London Math.

Soc. 35:601–607, 2003.

[30] H. Alzer. A superadditive property of hadamard’s gamma function. Abh.Math.Semin.Univ.Hambg. 79:11–23, 2009.

[31] H. Alzer, D. Karayannakis, H. M. Srivastava. Series representations for some mathematical

constants. J. Math. Anal. Appl. 320:145–162, 2006. Submitted by. H. M. Srivastava, 2. May2010.

[32] H. Alzer, S. Ruscheweyh. A subadditive property of the Gamma function. J. Math. Anal.

Appl. 285:564–577, 2003.[33] Horst Alzer. Some Gamma function inequalities. Math. Comp. [Mathematics of Computa-

tion] 60(201):337–346, 1993.[34] Horst Alzer, Jim Wells. Inequalities for the Polygamma functions. SIAM J. Math. Anal

29(6):1459–1466, 1998.

[35] Y. Amice, G. Christol, P. Robba, editors. Groupe d’etude d’analyse ultrametrique. 9e annee:1981/82. Fasc. 1. [Study Group on Ultrametric Analysis. 9th year: 1981/82. No. 1]. Institut

Henri Poincare, 115 pp., Paris, 1983. French.

[36] Y. Amice, G. Christol, P. Robba, editors. Groupe d’etude d’analyse ultrametrique. 9e annee:1981/82. Fasc. 3. [Study Group on Ultrametric Analysis. 9th year: 1981/82. No. 3]. Insti-

tut Henri Poincare, 125 pp, Paris, 1983. Conference: p-adic analysis and its applications;

September 6–10, Marseille, 1982; Edited by Y. Amice, In French.[37] Yvette Amice. Fonction γ p-adique associee a un caractere de Dirichlet. [p-adic γ function

associated with a Dirichlet character]. In Collection: Study Group on Ultrametric Analysis.

7th–8th years: 1979–1981 (Paris, 1979/1981) (French), Exp. No. 17, 11 pp, Paris, 1981.Secretariat Math. French.

[38] M. Amou. Irrationality results for values of certain q-functions. Proc. Jangjeon Math.

Soc.:27–36, 2000.[39] J. Anastassiadis. Definition des fonctions euleriennes par des equations fonctionnelles.

Gauthier-Villars, Paris, 1964.[40] Jean Anastassiadis. Fonctions semi-monotones et semi-convexes et solutions d’une equation

fonctionnelle. Bull. Sci. Math, 2e serie 76:148–160, 1952. French.

[41] Jean Anastassiadis. Sur les solutions logarithmiquement convexes ou concaves d’uneequation fonctionnelle. Bull. Sci. Math, 2e serie 81:78–87, 1952. French.

[42] Jean Anastassiadis. Une propriete de la fonction Gamma. Bull. Sci. Math, 2e serie 81:116–118, 1957. French.

[43] Jean Anastassiadis. Definitions fonctionnelles de la Fonction b(x, y). Bull. Sci. Math, 2e

serie 83:24–32, 1959. French.

[44] Jean Anastassiadis. Remarques sur les quelques equations fonctionnelles. C. R. Acad. Sci.Paris 250:2663–2665, 1960. French.

[45] Jean Anastassiadis. Sur les solutions de l’equation fonctionnelle f(x+ 1) = ϕ(x)f(x). C. R.Acad. Sci. Paris 253:2446–2447, 1961. French.

http://milanmerkle.com v.0.5 (October 2011)–Bibliography of Gamma function 3

[46] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen. Conformal invariants, inequalities andquasiconformal maps. Wiley, New York, 1997.

[47] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen. Topics in special functions. In Papers

on Analysis, a volume dedicated to Olli Martio, Report Univ. Jyvaskyla, volume 83, 5–26,2001.

[48] Greg W. Anderson. Logarithmic derivatives of Dirichlet l-functions and the periods of

Abelian varieties. Compositio Math. [Compositio Mathematica] 45(3):315–332, 1982.[49] Greg W. Anderson. Cyclotomy and an extension of the Taniyama group. Compositio Math.

[Compositio Mathematica] 57(2):153–217, 1986.[50] Greg W. Anderson. The hyperadelic Gamma function: A precis. In Collection: Galois

representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986), volume 12

of Adv. Stud. Pure Math, 1–19. North-Holland, Amsterdam-New York, 1987.[51] Greg W. Anderson. The hyperadelic Gamma function. Invent. Math. [Inventiones Mathe-

maticae] 95(1):63–131, 1989.

[52] Greg W. Anderson. Normalization of the hyperadelic Gamma function. In Collection: Galoisgroups over Q (Berkeley, CA, 1987), volume 16 of Math. Sci. Res. Inst. Publ, 1–31. Springer,

New York-Berlin, 1989.

[53] G.E. Andrews. q-Series: Their development and applications in Analysis, Number theory,Combinatorics, Physics and Computer algebra. Chelsea, USA, 1986. reprinted 1991.

[54] G.E. Andrews, R. Askey. Another q-extension of the Beta function. Proc. A.M.S. 81:97–100,

1981.[55] G.E. Andrews, R. Askey. Another q-extension of the Beta function. Proc. AMS 81(1):97–

100, 1981.

[56] G.E. Andrews, R. Askey, R. Roy. Special functions. Cambridge Univ. Press, 1999.[57] P. Appell. Sur un classe de fonctions analogues aux fonctions Euleriennes etudiees par M.

Heine. C.R. Acad. Sci. Paris 89:841–844, 1979.[58] T. Artikis. Convex Densities and Self-Decomposability. Serdica [Serdica. Bulgaricae Math-

ematicae Publicationes] 9(3):326–329, 1983.

[59] E. Artin. Einfuhrung in die theorie der Gammafunktion. B.G. Teubner, Berlin-Leipzig, 1931.[60] Emil Artin. The Gamma Function. Holt, Rinehart and Winston, New York, 1964. Transla-

tion from the German Original of 1931.

[61] R. Askey. A q-extension of Cauchy’s form of the Beta integral. Quart. J. Math. Oxford Ser.(2) 32(127):255–266, 1981.

[62] Richard Askey. The q-Gamma and qz-Beta functions. Applicable Anal. [Applicable Analysis.

An International Journal] 8(2):125–141, 1978–1979.[63] Richard Askey. Ramanujan’s extensions of the Gamma and Beta functions. Amer. Math.

Monthly [The American Mathematical Monthly] 87(5):346–359, 1980.

[64] J. P. Asselin de Beauville. Estimation non parametrique de la densite et du mode exem-ple de la distribution Gamma. Rev. Statist. Appl. [Revue de Statistique Appliquee. Centred’Enseignement et de Recherche de Statistique Appliquee] 26(3):47–70, 1978. French.

[65] A. C. Atkinson. The simulation of generalized inverse Gaussian and hyperbolic random

variables. SIAM J. Sci. Statist. Comput. [Society for Industrial and Applied Mathematics.

Journal on Scientific and Statistical Computing] 3(4):502–515, 1982.[66] A. M. Awad, M. M. Azzam, M. A. Hamdan. Power series mixture of Gamma distribution.

Iraqi J. Sci. [Iraqi Journal of Science] 22(1):108–121, 1981.

[67] Adnan M. Awad. On noncentral distribution and approximations to distributions of therandomized Gamma type. Dirasat Res. J. Natur. Sci. [Dirasat. A Research Journal. Natural

Sciences] 2(1):17–34, 1975.[68] K.-I. Babenko. Addendum: “The maximum principle for the Euler-Tricomi equation”. Dokl.

Akad. Nauk SSSR [Doklady Akademii Nauk SSSR] 287(3):520, 1986.

[69] M. Badiale. Some characterization of the q-gamma function by functional equations I. Atti

Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. (8) 74(1):7–11, 1983.[70] M. Badiale. Some characterization of the q-gamma function by functional equations II. Atti

Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. (8) 74(2):49–54, 1983.[71] Marino Badiale. Some characterisation of the q-Gamma function by functional equations. I.

Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur 8, 1983.

[72] Marino Badiale. Some characterization of the q-Gamma function by functional equations.II. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur 8, 1983.


[73] Marino Badiale, Francis J. Sullivan. A note on Badiale’s characterization of the q-Gammafunctions. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. [Atti della Accademia

Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisic] 8, 1984.

[74] D.H. Bailey. Numerical results on the transcendence of constants involving π, e, and Euler’sconstant. Math. Comp. 50(181):275–281, 1988.

[75] W.N. Bailey. Generalized hypergeometric series. Cambridge Univ. Press, 1935. In series:

Cambridge Tracts in Mathematics (Cambridge Math. Tract.), No. 32.[76] Francesco Baldassarri. Higher p-adic Gamma functions and Dwork cohomology. Asterisque

4(119–120):111–127, 1984. P -adic cohomology.[77] S.B. Bank. Some results on the Gamma function and other hypertranscendental functions.

Proc. Royal Soc. Edinb. Sect. A 79(3–4):335–341, 1977/78.

[78] Steven B. Bank. A note on zeros of differential polynomials in the Gamma function.Comment. Math. Univ. St. Paul. [Commentarii Mathematici Universitatis Sancti Pauli]

32(1):77–84, 1983.

[79] Steven B. Bank, Robert P. Kaufman. On the Gamma function and the Nevanlinna char-acteristic. Analysis [Analysis. International Journal of Analysis and its Application] 6(2–

3):115–133, 1986.

[80] G. Bankovi. A decomposition-rejection technique for generating exponential random vari-ables. Math. Proc. of the Hung. Acad. of Science, Ser. A 9:573–581, 1964.

[81] G. Bankovi. A note on the generation of Beta distributed and Gamma distributed random

variables. Math. Proc. of the Hung. Acad. of Science, Ser. A 9:555–562, 1964.[82] Shaul K. Bar Lev, Benjamin Reiser. A note on maximum conditional likelihood estimation

for the Gamma distribution. Sankhya Ser. B [Sankhya (Statistics). The Indian Journal of

Statistics. Series B] 45(2):300–302, 1983.[83] Mahmud Barjaktarevic. O rjesenjima jedne funkcionalne jednacine (On Solutions of a Func-

tional Equation). Glasnik Mat. Fiz. Astronom 8:297–300, 1953. Serbian.[84] E.W. Barnes. The theory of the G-function. Quart. J. Math. 31:264–314, 1899.

[85] E.W. Barnes. The theory of the double Gamma function. Philos. Trans. Roy. Soc. London

Ser. A 196:265–388, 1901.[86] E.W. Barnes. On the theory of the multiple Gamma-function. Cambr. Trans. 19:374–425,

1904.

[87] Daniel Barsky. On Morita’s p-adic γ-function. In Collection: Groupe d’Etude d’AnalyseUltrametrique, 5e annee (1977/78), Exp. No. 3, 6 pp., Paris, 1978. Secretariat Math.

[88] Daniel Barsky. On Morita’s p-adic Gamma function. Math. Proc. Cambridge Philos. Soc.

[Mathematical Proceedings of the Cambridge Philosophical Society] 89(1):23–27, 1981.[89] J. Bastero, F. Galve, A. Pena, M. Romano. Inequalities for the Gamma function and esti-

mates for the volume of sections of bnp . Proc. A.M.S. 130(1):183–192, 2001.

[90] G. Bastien, M. Rogalski. Convexite, complete monotonie et inegalites sur les fonctions Zetaet Gamma sur les fonctions des operateurs de Baskarov et sur des fonctions arithmetiques.

Canad. J. Math. 54(5):916–944, 2002.

[91] Jesus Basulto. Square root versus logarithmic transformation of a Gamma distribution. Tra-bajos Estadist. Investigacion Oper. [Trabajos de Estadistica y de Investigacion Operativa]

29(3):83–88, 1978. Spanish.[92] H. Bateman, A. Erdelyi. Higher transcendental functions. McGraw-Hill, New York, 1955.

[93] N. Batir. An interesting double inequality for Euler’s Gamma function. J. Ineq. Pure Appl.

Math. 5(4), 2004.[94] N. Batir. Some new inequalities for Gamma and Polygamma functions. J. Ineq. Pure Appl.

Math. 6(4), 2005.

[95] N. Batir. Sharp inequalities for factorial n. RGMIA Research Report Collection 9(3), 2006.[96] N. Batir. Some Gamma function inequalities. RGMIA Research Report Collection 9(3),

2006.

[97] E.F. Beckenbach, R. Bellman. Inequalities. Springer Verlag, 4th edition, 1983. first ed., 1961.[98] P. J. Becker, J. J. J. Roux. A bivariate extension of the Gamma distribution. South African

Statist. J. [South African Statistical Journal] 15(1):1–12, 1981.

[99] John Beebee. Exact covering systems and the Gauss-Legendre multiplication formula forthe Gamma function. Proc. Amer. Math. Soc. [Proceedings of the American MathematicalSociety] 120(4):1061–1065, 1994.


[100] P.R. Beesack. Inequalities for absolute moments of a distribution: From Laplace to vonMises. J. Math. Anal. Appl. 98:435–457, 1984.

[101] M. I. Beg. On estimation of scale parameter in Gamma distribution. IAPQR Trans. [IAPQR

Transactions. Journal of the Indian Association for Productivity, Quality and Reliability]12(1):83–85, 1987.

[102] Bradley M. Bell. Generalized Gamma parameter estimation and moment evaluation. Comm.

Statist. Theory Methods [Communications in Statistics. Theory and Methods] 17(2):507–517, 1988.

[103] M. Bencze. New generalization of the convexity (concavity). Octogon Math. M. 10(1):43–50,2002.

[104] M. Bencze. Special inequalities for increasing and convex functions. Octogon Math. M.

10(1):296–321, 2002.[105] M. Bencze. An inequality for convex functions. Octogon Math. M. 11(2):504–507, 2003.

[106] M. Bencze. Inequalities for Gamma function. Octogon Math. M. 13(2):982–984, 2005.

[107] M. Bencze, D.M. Batinetu-Giurgiu. Inequalities for Gamma function. Octogon Math. Mag.10(1):233–246, 2002.

[108] M. Bencze, C. J. Zhao. The lambda-multiplicatively convex (concave) version of Hadamard’s

inequality. Octogon Math. M. 10(1):93–102, 2002.[109] L. Bendersky. Sur la fonction Gamma generalisee. Acta Math. 61:263–322, 1933.

[110] C. Berg. Integral representation of some functions related to the Gamma function. Mediterr.

J. Math. 1:433–439, 2004. Submitted by. Christian Berg, 4. May 2010.[111] C. Berg. On a generalized Gamma convolution related to the q-calculus. In M. E. H. Ismail,

Erik Koelink, editors, Developments of Mathematics, ”Theory and Applications of Special

Functions”. A volume dedicated to Mizan Rahman., volume 13, 61–76. Springer Science+Business Media, Inc., Berlin, 2005. Submitted by. Christian Berg, 4. May 2010, ISBN: 0-

387-24231-7.[112] C. Berg, G. Forst. Potential theory on locally compact Abelian groups. Springer-Verlag,

Berlin, 1975.

[113] C. Berg, H. L. Pedersen. A Pick function related to the sequence of volumes of the unitball in n-space. Submitted by. Christian Berg, 4. May 2010, (ArXiv 0912.2185) (Manuscript

from December).

[114] C. Berg, H. L. Pedersen. A completely monotone function related to the Gamma function.J. Comput. Appl. Math. 133:219–230, 2001. Submitted by. Christian Berg, 4. May 2010.

[115] C. Berg, H. L. Pedersen. Pick functions related to the Gamma function. Rocky Mountain

Journal of Mathematics 32(2):507–525, 2002. Submitted by. Christian Berg, 4. May 2010.[116] C. Berg, H. L. Pedersen. The Chen-Rubin conjecture in a continuous setting. Methods and

Applications of Analysis 13(1):63–88, 2006. Submitted by. Christian Berg, 4. May 2010.

[117] C. Berg, H. L. Pedersen. Convexity of the median in the Gamma distribution. Ark. Mat.46:1–6, 2008. Submitted by. Christian Berg, 4. May 2010.

[118] A. Berger. Sur quelques applications de la fonction Gamma a la theorie des nombres. Akad.Afhandt., Uppsala, 1880.

[119] Mark Berman. The maximum likelihood estimators of the parameters of the Gamma distri-

bution are always positively biased. Comm. Statist. A. Theory Methods [Communicationsin Statistics. A. Theory and Methods] 10(7):693–697, 1981.

[120] B.C. Berndt. The Gamma function and the Hurwitz Zeta function. Amer. Math. Monthly

92:126–130, 1985.[121] B.C. Berndt, J. Sohn. Asymptotic formulas for two continued fractions in Ramanujan’s lost

notebook. J. London Math. Soc. 2(65):271–284, 2002.[122] Bruce C. Berndt. Chapter 8 of Ramanujan’s 2nd notebook. J. Reine Angew. Math. [Journal

fur die Reine und Angewandte Mathematik] 338:1–55, 1983.

[123] Bruce C. Berndt. The Gamma function and the Hurwitz Zeta-function. Amer. Math.

Monthly [The American Mathematical Monthly] 92(2):126–130, 1985.[124] Bruce C. Berndt. Ramanujan’s notebooks. Part I. Springer-Verlag, x+357 pp, $ 54.00., New

York-Berlin, 1985. With a foreword by S. Chandrasekhar.[125] Bruce C. Berndt, Robert L. Lamphere, B. M. Wilson. Chapter 12 of Ramanujan’s 2nd

notebook: Continued fractions. Rocky Mountain J. Math. [The Rocky Mountain Journal of

Mathematics] 15(2):235–310, 1985. Number theory (Winnipeg, Man, 1983).


[126] M. V. Berry. Infinitely many Stokes smoothings in the Gamma function. Proc. Roy. Soc.London Ser. A [Proceedings of the Royal Society. London. Series A. Mathematical and

Physical Sciences] 434(1891), 1991.

[127] Manjul Bhargava. The factorial function and generalizations. American MathematicalMonthly:783–799, 2000.

[128] Manish C. Bhattacharjee, N. Krishnaji. DFR and other heavy tail properties in modelling

the distribution of land and some alternative measures of inequality. In Collection: Statistics:Applications and new directions (Calcutta, 1981), 100–115, Calcutta, 1984. Indian Statist.

Inst.[129] Yang Bicheng. On a new extension of Hilbert’s double series theorem and applications. Int.

J. Appl. Math. Sci. 2(2):230–239, 2005.

[130] Yang Bicheng, Gao Mingzhe. On a best value of Hardy-Hilbert’s inequality. Advances inMath. 26(2):159–164, 1997.

[131] J. D. Biggins, N. H. Bingham. Near-constancy phenomena in branching processes. Math.

Proc. Cambridge Philos. Soc. [Mathematical Proceedings of the Cambridge PhilosophicalSociety] 110(3):545–558, 1991.

[132] D.M. Bloom, G. Booth. Problem 10521. Amer. Math. Monthly 103:347, 1996.

[133] C. Blyth, P.K. Pathak. A note on easy proofs of Stirling’s theorem. Amer. Math. Monthly93:376–379, 1986.

[134] Jan Bohman, Carl Erik Froberg. The γ-function revisited: Power series expansions and real

imaginary zero lines. Math. Comp. [Mathematics of Computation] 58(197):315–322, 1992.[135] Salvatore Bologna. Some families of distributions on finite intervals. Statistica (Bologna)

43(2):301–317, 1983. Italian.

[136] C.M. Bonan-Hamada, W.B. Jones. Stieltjes continued fractions for Polygamma functions,speed of convergence. J. Comp. Appl. Math. 179(1–2):47–55, 2005.

[137] L. Bondesson. On infinite divisibility of powers of a Gamma variable. Scand. ActuarialJ.:48–61, 1978.

[138] L. Bondesson. Generalized Gamma convolutions and related classes of distributions and

densities, volume 76 of Lecture Notes in Statistics. Springer Verlag, New York, 1992.[139] Lennart Bondesson. Correction and addendum to: “Classes of infinitely divisible distribu-

tions and densities”. Z. Wahrsch. Verw. Gebiete [Zeitschrift fur Wahrscheinlichkeitstheorie

und Verwandte Gebiete] 59(2):277, 1982.[140] C. Borelli Forti, I. Fenyo. On a generalization of Gamma functional equation. C. R. Math.

Rep. Acad. Sci. Canada [La Societe Royale du Canada.-L’Academie des Sciences. Comptes

Rendus Mathematiques. (Mathematical Reports)], 1982.[141] C. Borelli Forti, I. Fenyo. On a generalization of Gamma functional equation. C.R. Math.

Rep. Acad. Sci. Canada 4(5):261–265, 1982.

[142] David Borwein, Jonathan M. Borwein. On an intriguing integral and some series relatedto ζ(4). Proc. Amer. Math. Soc. [Proceedings of the American Mathematical Society]123(4):1191–1198, 1995.

[143] J. M. Borwein, P. B. Borwein. On the complexity of familiar functions and numbers. SIAM

Rev. [SIAM Review. A Publication of the Society for Industrial and Applied Mathematics]

30(4):589–601, 1988.[144] J. M. Borwein, I. J. Zucker. Fast evaluation of the Gamma function for small rational

fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal. [IMA

Journal of Numerical Analysis] 12(4):519–526, 1992.[145] J.M. Borwein, I.J. Zucker. Fast evaluation of the Gamma function for small rational fractions

using complete elliptic integrals of the first kind. IMA J. of Num. Analysis 12:519–526, 1992.[146] Peter B. Borwein. Reduced complexity evaluation of hypergeometric functions. J. Approx.

Theory [Journal of Approximation Theory] 50(3):193–199, 1987.

[147] Michael Boshernitzan. Discrete “orders of infinity”. Amer. J. Math. [American Journal of

Mathematics] 106(5):1147–1198, 1984.[148] N. Bourbaki. Fonctions d’une variable reelle. Hermann, Paris, 1951.

[149] Jean Francois Boutot, Lawrence Breen, Paul Gerardin, Jean Giraud, Jean Pierre Labesse,James Stuart Milne, Christophe Soule. Varietes de Shimura et fonctions L. [Shimura vari-

eties and L-functions], volume 6 of Publications Mathematiques de l’Universite Paris VII

[Mathematical Publications of the University of Paris VII]. Universite de Paris VII, U.E.R.de Mathematiques, 178 pp, 60 F., Paris, 1979. French.


[150] K. O. Bowman, L. R. Shenton. Sums of powers of binomial coefficients. Comm. Statist. Sim-ulation Comput. [Communications in Statistics. Simulation and Computation] 16(4):1189–

1207, 1987.

[151] K. O. Bowman, L. R. Shenton. Properties of estimators for the Gamma distribution., vol-ume 89 of Statistics: Textbooks and Monographs. Marcel Dekker, Inc. xvi+268 pp, $65.00.,

New York, 1988. With a contribution by Y. C. Patel.

[152] A. V. Boyd. Gurland’s inequality for the Gamma function. Skand. Aktuarietidiskr 43:134–135, 1960.

[153] D.W. Boyd. A p-adic study of the partial sums of the harmonic series. Experimental Math.3(4):287–302, 1994.

[154] P. Bracken. Properties of certain sequences related to Stirling’s approximation for the

Gamma function. Exp. Mathematicae 21:171–178, 2003.[155] Gudrun Brattstrom, Stephen Lichtenbaum. Jacobi-Sum Hecke characters of imaginary qua-

dratic fields. Compositio Math. [Compositio Mathematica] 53(3):277–302, 1984.

[156] Lee Brekke, Peter G. O. Freund, Ezer Melzer, Mark Olson. Adelic string n-point amplitudes.Phys. Lett. B [Physics Letters. B] 216(1–2):53–58, 1989.

[157] R.P. Brent. Ramanujan and Euler’s constant. In Math. Comp. 1943-1993: a half-century of

comp. math. (Vancouver, BC, 1993), 541–545. Proc. Sympos. Appl. Math. 48, AMS, 1994.[158] G. Brunel. Monographie de la fonction Gamma. Mem. de Bordeaux 3(3):1–184, 1886.

[159] H. Burkhardt. Zur Theorie der Gammafunktion, besonders uber ihre analytische Darstellung

fur große positive werte des Arguments. Sitzungsber. math.-phys. Kl. d. Kgl. BayerischenAkad. Wiss. Munchen:383–396, 1913.

[160] P. J. Bushell. On a generalization of Barton’s integral and related integrals of complete

elliptic integrals. Math. Proc. Cambridge Philos. Soc. [Mathematical Proceedings of theCambridge Philosophical Society] 101(1):1–5, 1987.

[161] Joaquin Bustoz, Mourad E. H. Ismail. On Gamma function inequalities. Math. Comp. [Math-ematics of Computation] 47(176):659–667, 1986.

[162] P.L. Butzer, M. Hauss. On Stirling functions of the second kind. Stud. Appl. Math. 84(1):71–

91, 1991.[163] R. Campbell. Les integrales Euleriennes et leurs applications. Dunod, Paris, 1966.

[164] A. Cannizzo. Analytic Kernels for integral representations of solutions of a functional equa-

tion. Boll. Un. Mat. Ital. B (7) [Unione Matematica Italiana. Bollettino. B. Serie VII]8(4):929–946, 1994. English.

[165] B.C. Carlson. Special functions of applied mathematics. Academic Press, 1977.

[166] M. Carmignani, G. Puleo, A. Tortorici Macaluso. Calculating to high precision the Euler-Mascheroni constant and generalized harmonic series. First applications to the calculation

of. Atti Accad. Sci. Lett. Arti Palermo Parte I 4, 1980–1981. Italian.

[167] Marta Carmignani, Adele Tortorici Macaluso. Computation of the special functionsγ(x), log γ(x), β(x, y), erf (x), erfc (x) to a high degree of. Atti Accad. Sci. Lett. ArtiPalermo Ser. (5) [Atti della Accademia di Scienze Lettere e Arti di Palermo] 2(1):7–25(1985), 1981–1982.

[168] Luigi Castoldi. The derivative of the Gamma function for positive integers as values of the

variables. Rend. Sem. Fac. Sci. Univ. Cagliari [Rendiconti del Seminario della Facolta diScienze della Universita di Cagliari] 48(3–4):35, 1978. Italian.

[169] P. P. Chakrabarty. On certain inequalities connected with the Gamma function. Skand.

Aktuarietidiskr:20–23, 1969.[170] Der Shin Chang. A note on incomplete Beta function. Nanta Math. [Nanyang University.

Nanta Mathematica] 12(1):71–74, 1979.[171] Derek K. Chang. A note on a characterization of Gamma distributions. Utilitas Math. [Util-

itas Mathematica. A Canadian Journal of Applied Mathematics, Computer Science and

Statistics] 35:153–154, 1989.

[172] Bruce W. Char. On Stieltjes’ continued fraction for the Gamma function. Math. Comp.[Mathematics of Computation] 34(150):547–551, 1980.

[173] S. D. Chatterji. Asymptotic formulae derived from the central limit theorem. Confer. Sem.Mat. Univ. Bari [Conferenze del Seminario di Matematica dell’Universita di Bari] 37

pp(234), 1990.


[174] M. A. Chaudhry, A. Qadir, H. M. Srivastava, R. B. Paris. Extended hypergeometric andconfluent hypergeometric functions. Appl. Math. Comput. 159:589–602, 2004. Submitted

by. H. M. Srivastava, 2. May 2010.

[175] C. P. Chen, F. Qi, H. M. Srivastava. Some properties of functions related to the Gammaand Psi functions. Integral Transforms Spec. Funct. 21:153–164, 2010. Submitted by. H. M.

Srivastava, 2. May 2010.

[176] C. P. Chen, H. M. Srivastava. A class of two-sided inequalities involving the Psi andPolygamma functions. Integral Transforms Spec. Funct. 21:523–528, 2010. Submitted by.

H. M. Srivastava, 2. May 2010.[177] C. P. Chen, H. M. Srivastava. Some inequalities and monotonicity properties associated with

the Gamma and Psi functions and the Barnes G-function. Integral Transforms Spec. Funct.

21, 2010. Submitted by. H. M. Srivastava, 2. May 2010, (to appear).[178] C.P. Chen. Monotonicity and convexity for the Gamma function. J. Ineq. Pure Appl. Math.

6(4), 2005.

[179] C.P. Chen, F. Qi. The best lower and upper bounds of harmonic sequence. RGMIA ResearchReport Collection 6(2), 2003.

[180] Jeesen Chen, Herman Rubin. Bounds for the difference between median and mean of Gamma

and Poisson distributions. Statist. Probab. Lett. [Statistics and Probability Letters] 4(6):281–283, 1986.

[181] K. J. Chen, H. M. Srivastava. A generalization of two q-identities of Andrews. J. Combin.

Theory Ser. A 95:381–386, 2001. Submitted by. H. M. Srivastava, 2. May 2010.[182] K. Y. Chen, H. M. Srivastava. Some infinite series and functional relations that arose in the

context of fractional calculus. J. Math. Anal. Appl. 252:376–388, 2000. Submitted by. H.

M. Srivastava, 2. May 2010.[183] C. Chiccoli, S. Lorenzutta, G. Maino. A note on a Tricomi expansion for the generalized

exponential integral and related functions. Nuovo Cimento B (11) [Il Nuovo Cimento. B.Serie 11] 103(5):563–568, 1989.

[184] C. Chiccoli, S. Lorenzutta, G. Maino. A note on a Tricomi expansion for the generalized

exponential integral and related functions. Nuovo Cimento B (11) [Il Nuovo Cimento. B.Serie 11] 103(5):563–568, 1989.

[185] J. Choi, P. J. Anderson, H. M. Srivastava. Some q-extensions of the Apostol-Bernoulli and

the Apostol-Euler polynomials of order n, and the multiple Hurwitz Zeta function. Appl.Math. Comput. 199:723–737, 2008. Submitted by. H. M. Srivastava, 2. May 2010.

[186] J. Choi, P. J. Anderson, H. M. Srivastava. Carlitz’s q-Bernoulli and q-Euler numbers and

polynomials and a class of q-Hurwitz Zeta functions. Appl. Math. Comput. 215:1185–1208,2009. Submitted by. H. M. Srivastava, 2. May 2010.

[187] J. Choi, Y. J. Cho, H. M. Srivastava. Log-sine integrals involving series associated with the

Zeta function and Polylogarithms. Math. Scand. 105:199–217, 2009. Submitted by. H. M.Srivastava, 2. May 2010.

[188] J. Choi, J. Lee, H. M. Srivastava. A generalization of Wilf’s formula. Kodai Math. J. 26:44–48, 2003. Submitted by. H. M. Srivastava, 2. May 2010.

[189] J. Choi, J.R. Quine. Barnes’ approach of the multiple Gamma functions. J. Korean Math.

Soc. 29:127–140, 1992.[190] J. Choi, T.Y. Seo. The double Gamma function. East Asian Math. J. 13:159–174, 1997.

[191] J. Choi, T.Y. Seo. Integral formulas for Euler’s constant. Commun. Korean Math. Soc.

13(3):683–689, 1998.[192] J. Choi, H. M. Srivastava. An application of the theory of the double Gamma function.

Kyushu J. Math. 53:209–222, 1999. Submitted by. H. M. Srivastava, 2. May 2010.[193] J. Choi, H. M. Srivastava. Certain classes of series associated with the Zeta function and

multiple Gamma functions. J. Comput. Appl. Math. 118:87–109, 2000. Submitted by. H.

M. Srivastava, 2. May 2010.

[194] J. Choi, H. M. Srivastava. Evaluation of higher-order derivatives of the Gamma function.Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11:201–210, 2000. Submitted by. H. M.

Srivastava, 2. May 2010.[195] J. Choi, H. M. Srivastava. A certain family of series associated with the Zeta and related

functions. Hiroshima Math. J. 32:417–429, 2002. Submitted by. H. M. Srivastava, 2. May

2010.


[196] J. Choi, H. M. Srivastava. Explicit evaluation of Euler and related sums. Ramanujan J.10:51–70, 2005. Submitted by. H. M. Srivastava, 2. May 2010.

[197] J. Choi, H. M. Srivastava. A family of log-Gamma integrals and associated results. J. Math.

Anal. Appl. 303:436–449, 2005. Submitted by. H. M. Srivastava, 2. May 2010.[198] J. Choi, H.-M Srivastava. A note on a multiplication formula for the multiple Gamma

function Gamman. Italian J. Pure Appl. Math. 23:179–188, 2008.

[199] J. Choi, H. M. Srivastava. A note on a multiplication formula for the multiple Gammafunction gn. Italian J. Pure Appl. Math. 23:179–188, 2008. Submitted by. H. M. Srivastava,

2. May 2010.[200] J. Choi, H. M. Srivastava. Integral representations for the Gamma function, the Beta func-

tion, and the double Gamma function. Integral Transforms Spec. Funct. 20:859–869, 2009.

Submitted by. H. M. Srivastava, 2. May 2010.[201] J. Choi, H. M. Srivastava. Some applications of the Gamma and Polygamma functions

involving convolutions of the Rayleigh functions, multiple Euler sums and log-sine integrals.

Math. Nachr. 282:1709–1723, 2009. Submitted by. H. M. Srivastava, 2. May 2010.[202] J. Choi, H. M. Srivastava. Integral representations for the Euler-Mascheroni constant γ.

Integral Transforms Spec. Funct. 21:675–690, 2010. Submitted by. H. M. Srivastava, 2. May

2010.[203] J. Choi, H. M. Srivastava. Mathieu series and associated sums involving the Zeta function.

Comput. Math. Appl. 59:861–867, 2010. Submitted by. H. M. Srivastava, 2. May 2010.

[204] J. Choi, H. M. Srivastava, V. S. Adamchik. Multiple Gamma and related functions. Appl.Math. Comput. 134:515–533, 2003. Submitted by. H. M. Srivastava, 2. May 2010.

[205] J. Choi, H. M. Srivastava, Y. Kim. Applications of a certain family of hypergeometricsummation formulas associated with the Psi and Zeta functions. Comm. Korean Math. Soc.

16:319–332, 2001. Submitted by. H. M. Srivastava, 2. May 2010.

[206] J. Choi, H. M. Srivastava, T. Y. Seo, A. K. Rathie. Some families of infinite series. SoochowJ. Math. 25:209–219, 1999. Submitted by. H. M. Srivastava, 2. May 2010.

[207] J. Choi, H. M. Srivastava, N. Y. Zhang. Integrals involving a function associated with the

Euler-Maclaurin summation formula. Appl. Math. Comput. 93:101–116, 1998. Submittedby. H. M. Srivastava, 2. May 2010.

[208] J. Choi, H.M. Srivastava. An application of the theory of the double Gamma function.

Kyushu J. Math. 53(1):209–222, 1999.[209] K. P. Choi. On the medians of Gamma distributions and an equation of Ramanujan. Proc.

Amer. Math. Soc. [Proceedings of the American Mathematical Society] 121(1):245–251,

1994.[210] J. Choi et al. Series involving the Zeta function and multiple Gamma functions. Appl. Math.

Comp. 159:509–537, 2004.

[211] Gilles Christol. P -adic numbers and ultrametricity. In Collection: From number theory tophysics (Les Houches, 1989), 440–475. Springer, Berlin, 1992.

[212] J.T. Chu. A modified Wallis product and some applications. Amer. Math. Monthly 69:402–404, 1962.

[213] Walford I. E. Chukwu, Devendra Gupta. On mixing generalized Poisson with generalized

Gamma distribution. Metron 47(1–4):313–319, 1989.[214] E. Cinlar. Correction: “On a generalization of Gamma processes”. J. Appl. Probab. [Journal

of Applied Probability] 17(3):893, 1980.

[215] J. Cizek, E.R. Vrscay. Asymptotic estimation of the coefficients of the continued fractionrepresenting the Binet function. C.R. Math. Rep. Acad. Sci. Canada, IV 4:201–206, 1982.

[216] W. J. Cody. Performance evaluation of programs related to the real Gamma function. ACMTrans. Math. Software [Association for Computing Machinery. Transactions on Mathemat-ical Software] 17(1):46–54, 1991.

[217] A. Clifford Cohen, Betty Jones Whitten. Modified moment and maximum likelihood estima-

tors for parameters of the three-parameter Gamma distribution. Comm. Statist. B Simula-tion Comput. [Communications in Statistics. B. Simulation and Computation] 11(2):197–

216, 1982.[218] R. F. Coleman. Anderson-Ihara theory: Gauss sums and circular units. In Collection: Alge-

braic number theory, volume 17 of Adv. Stud. Pure Math, 55–72. Academic Press, Boston,

MA, 1989.


[219] Robert F. Coleman. The Gross-Koblitz formula. In Collection: Galois representations andarithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986), volume 12 of Adv. Stud. Pure

Math, 21–52, Amsterdam-New York, 1987. North-Holland.

[220] Pierre Colmez. Periodes de varietes Abeliennes a multiplication complexe et derivees defonctions l d’Artin en s = 0. [Periods of Abelian varieties with complex multiplication and

derivatives of Artin l-functions at s = 0]. C. R. Acad. Sci. Paris Ser. I Math. [Comptes

Rendus des Seances de l’Academie des Sciences. Serie I. Mathematique] 309(3):139–142,1989. French.

[221] Pierre Colmez. Periodes des varietes Abeliennes a multiplication complexe. [Periods ofAbelian varieties with complex multiplication]. Ann. of Math. (2) [Annals of Mathemat-

ics. Second Series] 138(3):625–683, 1993. French.

[222] Matthijs Coster. Generalisation of a congruence of Gauss. J. Number Theory [Journal ofNumber Theory] 29(3):300–310, 1988.

[223] Larry H. Crow. Confidence interval procedures for the Weibull process with applications to

reliability growth. Technometrics [Technometrics. A Journal of Statistics for the Physical,Chemical and Engineering Sciences] 24(1):67–72, 1982.

[224] B. Sekeroglu, H. M. Srivastava, F. Tasdelen. Some properties of q-biorthogonal polynomials.

J. Math. Anal. Appl. 326:896–907, 2006. Submitted by. H. M. Srivastava, 2. May 2010.[225] L. Cupello. An inequality involving the incomplete Gamma function. Riv. Mat. Sci. Econom.

Social. [Rivista di Matematica per le Scienze Economiche e Sociali] 15(2):55–62, 1992.

Italian.[226] Dragos Cvetkovic. O sumiranju redova ciji je opsti clan racionalna funkcija indeksa

sumiranja. In Radovan Janic, editor, Uvodj enje mladih u naucni rad VI, volume 41 of

Matematicka biblioteka, 157–178. Zavod za izdavanje udzbenika Socijalisticke Republike Sr-bije, Beograd, 1969. Serbian.

[227] Ram C. Dahiya, John Gurland. Estimating the parameters of a Gamma distribution. Tra-bajos Estadist. Investigacion Oper. [Trabajos de Estadistica y de Investigacion Operativa]

29(2):81–87, 1978.

[228] S. R. Dalal, C. L. Mallows. When should one stop testing software? J. Amer. Statist. Assoc.[Journal of the American Statistical Association] 83(403):872–879, 1988.

[229] Anirban Das Gupta. Simultaneous estimation in the multiparameter Gamma distribution

under weighted quadratic losses. Ann. Statist. [The Annals of Statistics] 14(1):206–219,1986.

[230] Joseph W. Dauben, editor. Mathematical Perspectives. Academic Press, Inc. [Harcourt Brace

Jovanovich, Publishers], 1981, viii+272 pp. (1 plate), $34.00., New York-London, 1981.Essays on mathematics and its historical development; Papers in honor of Kurt-Reinhard

Biermann on the occasion of his 60th birthday.

[231] H.T. Davis. An extension to Polygamma functions of a theorem of gauss. Bull. Amer. M.S.41:243–247, 1935.

[232] Philip J. Davis. Leonhard Euler’s integral: A historical profile of the Gamma function.Amer. Math. Monthly 66:849–869, 1959.

[233] D. E. Daykin, C. J. Eliezer. Generalization of Holder’s and Minkowski’s inequalities. Proc.

Camb. Phil. Soc 64:1023–1027, 1968.[234] P. J. de Doelder. On the Clausen integral cl2(θ) and a related integral. J. Comput. Appl.

Math. [Journal of Computational and Applied Mathematics] 11(3):325–330, 1984.

[235] Jesus de la Cal, Francisco Luquin Martinez. Approximating Szasz and Gamma operators byBaskakov operators. J. Math. Anal. Appl. [Journal of Mathematical Analysis and Applica-

tions] 184(3):585–593, 1994.[236] S. Salinas de Romero, H. M. Srivastava. Some families of infinite sums derived by means of

fractional calculus. East Asian Math. J. 17:135–146, 2001. Submitted by. H. M. Srivastava,

2. May 2010.

[237] B. Dennis, G. P. Patil. The Gamma distribution and weighted multimodal Gamma distri-butions as models of population abundance. Math. Biosci. [Mathematical Biosciences. An

International Journal] 68(2):187–212, 1984.[238] G. M. D’Este. A Morgenstern-type bivariate Gamma distribution. Biometrika 68(1):339–

340, 1981.

[239] P. Diaconis, D. Freedman. An elementary proof of Stirling’s formula. Amer. Math. Monthly93:123–125, 1986.


[240] J. Diamond. The p-adic log Gamma function and p-adic Euler constants. Trans. Amer.Math. Soc. 233:321–337, 1977.

[241] Jack Diamond. On the values of p-adic l-functions at positive integers. Acta Arith. [Polska

Akademia Nauk. Instytut Matematyczny. Acta Arithmetica] 35(3):223–237, 1979.[242] Jack Diamond. The p-adic Gamma measures. Proc. Amer. Math. Soc. [Proceedings of the

American Mathematical Society] 75(2):211–217, 1979.

[243] Jack Diamond. Hypergeometric series with a p-adic variable. Pacific J. Math. [Pacific Jour-nal of Mathematics] 94(2):265–276, 1981.

[244] Jack Diamond. p-adic Gamma functions and their applications. In Collection: Number the-ory (New York, 1982), volume 1052 of Lecture Notes in Math, 168–175, Berlin-New York,

1984. Springer.

[245] T. J. DiCiccio. Approximate inference for the generalized Gamma distribution. Technomet-rics. A Journal of Statistics for the Physical, Chemical and Engineering Sciences 29(1):33–

40, 1987.

[246] A.R. DiDonato, A.H. Morris, Jr. Computation of the incomplete Gamma functions. ACMTrans. Math. Software 12:377–393, 1986.

[247] Hongming Ding, Kenneth I. Gross. Operator-valued Bessel functions on Jordan algebras.

J. Reine Angew. Math. [Journal fur die Reine und Angewandte Mathematik] 435:157–196,1993.

[248] A. Dinghas. Zur Theorie der Gammafunktion. Math. Phys. Semesterber 6:245–252, 1958/59.

[249] Alexander Dinghas. Zur Definition der Gammafunktion durch die Eulersche Funktionalgle-ichung. Anal. Stiint. Univ. ”Al.I.Cuza”. Iasi (ser.noua), sect. I a. Mat, fasc.1 19:31–35,

1973. German.

[250] G.L. Dirichlet. Werke, volume 1. G. Reimer, pp. 375-380, Berlin, 1897. see also C.R. Acad.Sci. Paris 8(1839), 156–160.

[251] Ulhas J. Dixit. Estimation of parameters of the Gamma distribution in the presence of out-liers. Comm. Statist. Theory Methods [Communications in Statistics. Theory and Methods]

18(8):3071–3085, 1989.

[252] Ulhas J. Dixit. On the estimation of the power of the scale parameter of the Gamma dis-tribution in the presence of outliers. Comm. Statist. Theory Methods [Communications in

Statistics. Theory and Methods] 20(4):1315–1328, 1991.

[253] J.L. Doob. Classical potential theory and its probabilistic counterpart. Springer–Verlag, NewYork, 1984.

[254] M. R. Dotsenko. On an Integral Transform with Wright’s Hypergeometric Function. Mat.

Fiz. Nelinein. Mekh. [Akademiya Nauk Ukrainy. Institut Matematiki. MatematicheskayaFizika i Nelineinaya Mekhanika] 18(52):47–52, 1993. Russian.

[255] S.S. Dragomir, R.P. Agarwal, N.S. Barnett. Inequalities for Beta and Gamma functions

via some classical and new inequalities. RGMIA Research Report Collection (Australia)2(3):283–333, 1999.

[256] S.S. Dragomir, J. Pecaric, J. Sandor. A note on the Jensen-Hadamard inequality. l’Anal.Numer. Th. Approx. (Cluj) 19:29–34, 1990.

[257] S.D. Dubey. Statistical determination of certain mathematical constants and functions using

computers. J. Assoc. Comput. Mach. 13:511–525, 1966.[258] Serge Dubuc. An approximation of the Gamma function. J. Math. Anal. Appl. [Journal of

Mathematical Analysis and Applications] 146(2):461–468, 1990.

[259] Helmut P. Dudel, Charles E. Hall, Jr., Siegfried H. Lehnigk. Parameter estimation algorithmsfor the Hyper-Gamma distribution class. Comm. Statist. Simulation Comput. [Communi-

cations in Statistics. Simulation and Computation] 18(3):1135–1153, 1989.[260] R. L. Duncan. A probability distribution associated with the Hurwitz Zeta function. Proc.

Amer. Math. Soc. [Proceedings of the American Mathematical Society] 99(4):757–759, 1987.

[261] Jacques Dutka. Wallis’s product, Brouncker’s continued fraction, and Leibniz’s series. Arch.

Hist. Exact Sci. [Archive for History of Exact Sciences] 26(2):115–126, 1982.[262] Jacques Dutka. On some Gamma function inequalities. SIAM J. Math. Anal. [SIAM Journal

on Mathematical Analysis] 16(1):180–185, 1985.[263] Anne Duval. Etude asymptotique d’une integrale analogue a la fonction “γ modifiee”. In

Collection: Differential equations and Pfaffian systems in the complex field, II, volume 1015

of Lecture Notes in Math, 50–63, Berlin-New York, 1983. Springer.


[264] Anne Duval. Etude asymptotique d’une integrale analogue a la fonction “γ modifiee”. InCollection: Differential equations and Pfaffian systems in the complex field, II, volume 1015

of Lecture Notes in Math., 50–63. Springer, Berlin-New York, 1983.

[265] H.B. Dwight. Tables of integrals and other mathematical data. Macmillan Comp., N.Y., 4thedition, 1961.

[266] Bernard Dwork. A note on the p-adic Gamma function. In Collection: Study group on

ultrametric analysis, 9th year: 1981/82, No. 3 (Marseille, 1982), Exp. No. J5, 10 pp.,Paris, 1983. Inst. Henri Poincare.

[267] Bernard Dwork. Generalized Hypergeometric Functions. Oxford Mathematical Monographs.The Clarendon Press, Oxford University Press, vi+188 pp, $63.00, New York, 1990. Oxford

Science Publications.

[268] K. O. Dzaparidze, M. S. Nikulin. The probability distributions of the Kolmogorov andOmega-squared statistics for continuous distributions with shift and location parameters.

Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) [Zapiski Nauchnykh

Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. SteklovaAkademii Nauk SSSR (LOMI)] 85:46–74, 238, 245, 1979. Investigations in the theory of

probability distributions, IV., Russian.

[269] A. Eagle. A simple theory of the Gamma function. Math. Gaz. 14:118–127, 1928.[270] Walther Eberl, Jr. A characterization of the Gamma distribution by optimal estimates. Teor.

Veroyatnost. i Primenen. [Akademiya Nauk SSSR. Teoriya Veroyatnostei i ee Primeneniya]

30(4):804–810, 1985. Russian.[271] Walther Eberl, Jr. Invariant sufficiency, equivariance and characterizations of the Gamma

distribution. In Collection: Contributions to stochastics, 101–122. Physica, Heidelberg,

1987.[272] Muhammad El Taha. MVU estimation in a shifted Gamma distribution with shape param-

eter a known integer. Comm. Statist. Simulation Comput. [Communications in Statistics.Simulation and Computation] 22(3):831–843, 1993.

[273] N. Elezoric, C. Giordano, J. Pecaric. Convexity and q-Gamma function. Rend. Circ. Mat.

Palermo, XLVIII 2:285–298, 1999.[274] N. Elezovic, C. Giordano, J. Pecaric. The best bound in Gautschi’s inequality. Math. Ineq.

Appl. 3:239–259, 2000.

[275] C. J. Eliezer. Problem 5798. Amer. Math. Monthly 78:549, 1971.[276] Mikihiko Endo. P -adic multiple Gamma functions. Comment. Math. Univ. St. Paul. [Com-

mentarii Mathematici Universitatis Sancti Pauli] 43(1):35–54, 1994.

[277] Jan Engel, Mynt Zijlstra. A characterization of the Gamma distribution by the negativebinomial distribution. J. Appl. Probab. [Journal of Applied Probability] 17(4):1138–1144,

1980.

[278] Leonhard Euler. Zur Theorie Komplexer Funktionen. [On the Theory of Complex Func-tions], volume 261 of Ostwalds Klassiker der Exakten Wissenschaften [Ostwald’s Classicsof the Exact Sciences]. Akademische Verlagsgesellschaft Geest & Portig K. G. 263 pp, MDN34.00., Leipzig, 1983. Edited by H. Wussing., German.

[279] R.J. Evans. Multidimensional Beta and Gamma integrals. In The Rademacher legacy to

mathematics (Univ. Park, PA, 1992), 341–357. Contemp. Math. 166, AMS, 1994.[280] R.J. Evans. Multidimensional q-Beta integrals. SIAM J. Math. Anal. 23(3):758–765, 1992.

[281] C.J. Everett. Inequalities for the Wallis Product. Math. Mag 43:30–33, 1970.

[282] M. A. Evgrafov. Series and integral representations. In Collection: Current problems ofmathematics. Fundamental directions, volume 13 of Itogi Nauki i Tekhniki, 5–92. Akad.

Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform, Moscow, 1986. Russian.[283] M.A. Evgrafov. Asymptotic estimates and entire functions. Gordon and Breach, New York,

1961. A.L. Shields, Transl.

[284] Jacques Faraut. Formule du binome generalisee. [The generalized binomial formula]. In Col-

lection: Harmonic analysis (Luxembourg, 1987), volume 1359 of Lecture Notes in Math,170–180, Berlin-New York, 1988. Springer. French.

[285] W. Feller. An introduction to Probability theory and its Applications, volume 2. Wiley, NewYork, 1966.

[286] W. Feller. A direct proof of Stirling’s formula. Amer. Math. Monthly 74:1223–1225, 1967.

Correction: ibid: vol. 75 (1968) , 518.


[287] Emil A. Fellmann. Uber Einige Mathematische Sujets im Briefwechsel Leonhard Eulers mitJohann Bernoulli. [On some mathematical subjects in the correspondence]. In Collection:

On the work of Leonhard Euler (Berlin, 1983), 39–66. Birkhauser, Basel-Boston, Mass,

1984. German.[288] I. Fenyo. The solution of a functional equation by Laplace transformation. In Collection:

Generalized functions and operational calculus (Proc. Conf, Varna, 1975), 97–100, Sofia,

1979. Bulgar. Acad. Sci.[289] Carlo Ferreri. On the hypergeometric birth process and some implications about the Gamma

distribution representation. J. Roy. Statist. Soc. Ser. B [Journal of the Royal StatisticalSociety. Series B. Methodological] 46(1):52–57, 1984.

[290] S. Finch. Mathematical constants. http://www.mathsoft.com/asolve/constant/constant.html.[291] Brian Fisher. On defining γ(−n) for n = 0, 1, 2, · · · . Rostock. Math. Kolloq. [Rostocker

Mathematisches Kolloquium](31):4–10, 1987.

[292] R. Floreani, L. Vinet. q-Gamma and q-Beta functions in quantum algebra representation

theory. J. Comput. Appl. Math. 68(1–2):57–68, 1996.[293] Tomlinson Fort. The Gamma function with varying difference interval. Amer. Math. Monthly

73(9):951–959, 1966.

[294] Arne Fransen. Addendum and corrigendum to: “High-precision values of the Gamma func-tion and of some related coefficients” [Math. Comp. 34 (1980), no. 1 553-566]. Math. Comp.

[Mathematics of Computation] 37(155):233–235, 1981.

[295] Arne Fransen, Staffan Wrigge. High-precision values of the Gamma function and of somerelated coefficients. Math. Comp. [Mathematics of Computation] 34(150):553–566, 1980.

[296] Arne Fransen, Staffan Wrigge. Calculation of the moments and the moment generating func-

tion for the reciprocal Gamma distribution. Math. Comp. [Mathematics of Computation]42(166):601–616, 1984.

[297] C. L. Frenzen. Error bounds via complete monotonicity for a uniform asymptotic expansion

of the Legendre function p−mn (cosh z). In Collection: Asymptotic and computational analysis

(Winnipeg, MB, 1989), volume 124 of Lecture Notes in Pure and Appl. Math, 587–599.Dekker, New York, 1990.

[298] C.L. Frenzen. Error bounds for asymptotic expansions of the ratio of two Gamma functions.

SIAM J. Math. Anal. 18:890–896, 1987.[299] Peter G. O. Freund. 2-d Adelic strings. In Collection: Proceedings of the XXth International

Conference on Differential Geometric Methods in Theoretical Physics, (New York, 1991),

volume 1–2, 780–783, River Edge, NJ, 1992. World Sci. Publishing.[300] Peter G. O. Freund, Anton V. Zabrodin. A hierarchic array of integrable models. J.

Math. Phys. [Journal of Mathematical Physics] 34(12):5832–5842, 1993. Paper gives a veryclear and comprehensive account of generalizations of the Gamma function termed Higher

Gamma functions, of which the Barnes’ G-function is the first, and their corresponding

q-generalizations.[301] F. Fricker. Einfuhrung in die Gitterpunktlehre. Birkhauser Verlag, 1982.

[302] Aime Fuchs, G. Letta. Un resultat elementaire de fiabilite. application a la formule de

Weierstrass sur la fonction Gamma. [An elementary reliability result. Application to theWeierstrass formula on the Gamma function]. In Collection: Seminaire de Probabilites,

XXV, volume 1485 of Lecture Notes in Math, 316–323, Berlin, 1991. Springer. French.

[303] B. Fuglede. A sharpening of Wielandt’s characterization of the Gamma function. Amer.Math. Monthly 115:845–850, 2008. Submitted by. Christian Berg, 4. May 2010.

[304] Functional equations and their applications., Kyoto, April 7–9 1983. Kyoto University, Re-

search Institute for Mathematical Sciences, pp. i–ii and 1–205. Symposium: Functionalequations and their applications. Proceedings of a symposium held at the Research Insti-

tute for Mathematical Sciences.[305] R. Funke. A nonlinear diffusion with Hyper-Gamma distribution. Stochastic Anal. Appl.

[Stochastic Analysis and Applications] 7(1):19–33, 1989.[306] Leslaw Gajek. A characterization and some properties of generalized Gamma distri-

bution. Zeszyty Nauk. Politech. Lodz. Mat. [Zeszyty Naukowe Politechniki Lodzkiej.

Matematyka](15):149–164, 1982.

[307] P. Gao. Some monotonicity properties of the q-Gamma function. RGMIA Research ReportCollection 8(3), 2005.


[308] P. Gao. A subadditive property of the Digamma function. RGMIA Research Report Collec-tion 8(3), 2005.

[309] Jose Garay de Pablo, Maria Jose Parrilla Hernandez. Calculating limits by a Gauss formula

for the incomplete Gamma function. In Collection: Proceedings of the ninth conferenceof Portuguese and Spanish mathematicians 1, Acta Salmanticensia. Ciencias, 46, 285–288,

Salamanca, 1982. Univ. Salamanca.

[310] Jose Garay de Pablo, Maria Jose Parrilla Hernandez. Calculating limits by a Gauss formulafor the incomplete Gamma function. In Collection: Proceedings of the ninth conference of

Portuguese and Spanish mathematicians, 1 (Salamanca, 1982), volume 46 of Acta Salman-ticensia. Ciencias, 285–288. Univ. Salamanca, Salamanca, 1982.

[311] George Gasper, Mizan Rahman. Basic Hypergeometric Series., volume 35 of Encyclopedia

of Mathematics and its Applications. Cambridge University Press, xx+287 pp, $59.50.,Cambridge, 1990. With a foreword by Richard Askey.

[312] Luigi Gatteschi. On some orthogonal polynomial integrals. Math. Comp. [Mathematics of

Computation] 35(152):1291–1298, 1980.[313] W. Gautschi. Some elementary inequalities relating to the Gamma and incomplete Gamma

function. Journal of Mathematics and Physics 38(1):77–81, 1959.

[314] W. Gautschi. A harmonic mean inequality for the Gamma function. SIAM J. Math. Anal5(2):278–281, 1974.

[315] W. Gautschi. Some mean value inequalities for the Gamma function. SIAM J. Math. Anal

5(2):282–292, 1974.[316] W. Gautschi. Algorithm 542-Incomplete Gamma function. ACM Trans. Math. Software

5:482–489, 1979.

[317] W. Gautschi. A computational procedure for incomplete Gamma function. ACM Trans.Math. Software 5:466–481, 1979.

[318] W. Gautschi. Un procedimento di calcolo per le funzioni Gamma incomplete. Rend. Semin.Mat. Torino 37:1–9, 1979.

[319] Walter Gautschi. Polynomials orthogonal with respect to the reciprocal Gamma function.

BIT [Nordisk Tidskrift for Informationsbehandling (BIT)] 22(3):387–389, 1982.[320] Walter Gautschi. On the convergence behavior of continued fractions with real elements.

Math. Comp. [Mathematics of Computation] 40(161):337–342, 1983.

[321] Walter Gautschi. Summation of slowly convergent series. In Collection: Numerical analysisand mathematical modelling, volume 29 of Banach Center Publ., 7–18, 1994.

[322] P. Gavruta. A characterization of Gamma function. Inst. Politehn.-“Traian Vuia”

Timisoara. Lucrar. Sem. Mat. Fiz. [Institutului Politehnic “Traian Vuia” Timisoara. Lu-crarile Seminarului de Matematica si Fizica]:54–56, November 1985.

[323] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky. Generalized Euler Integrals and A-

hypergeometric functions. Adv. Math. 84:255–271, 1990.[324] A. Gertsch. Nombres harmoniques generalises. C.R. Acad. Sci. Paris, Serie I 324:7–10,

1997.[325] A.O. Ghelfond. Calculus of finite differences (Romanian). Ed. Tehnica, Bucuresti, 1956.

Translated from the Russian ed. of 1952.

[326] C. Giordano, A. Laforgia, J. Pecaric. Supplements to known inequalities for some specialfunctions. J. Math. Anal. Appl. 200(1):34–41, 1996.

[327] C. Giordano, A. Laforgia, J. Pecaric. Unified treatment of Gautschi-Kershan type inequali-

ties for the Gamma function. J. Comp. Appl. Math. 99:167–175, 1998.[328] J.W.L. Glaisher. On the history of Euler’s constant. Messenger Math. 1:25–30, 1872.

[329] J.W.L. Glaisher. On the product 11 · 22 · 33 . . . nn. Messenger Math. 2(7):43–47, 1877.[330] Leon Jay Gleser. The Gamma distribution as a mixture of exponential distributions. Amer.

Statist. [The American Statistician] 43(2):115–117, 1989.

[331] Maurice M. Godefroy. La fonction Gamma: theorie, histoire, bibliographie. Gauthier-Villars,

Paris, 1901. Theses presentees pour obtenir le titre de docteur de l’universite de Paris,French.

[332] Goran Gogic. Kurepina hipoteza o levom faktorijelu. Racunarstvo 1(2):41–45, 1992. Serbian.[333] D. Gokhale. On an inequality for Gamma function. Skand. Aktuarietidiskr:210–215, 1962.

[334] I.J. Good. Short proof of a conjecture of Dyson. J. Math. Phys. 11:1884, 1970.

[335] Louis Gordon. Bounds for the distribution of the generalized variance. Ann. Statist. [TheAnnals of Statistics] 17(4):1684–1692, 1989.


[336] Louis Gordon. A stochastic approach to the Gamma function. Amer. Math. Monthly [TheAmerican Mathematical Monthly] 101(9):858–865, 1994.

[337] David Goss. The γ-function in the arithmetic of function fields. Duke Math. J. [Duke Math-

ematical Journal] 56(1):163–191, 1988.[338] H. W. Gould, H. M. Srivastava. Some combinatorial identities associated with the Vander-

monde convolution. Appl. Math. Comput. 84:97–102, 1997. Submitted by. H. M. Srivastava,

2. May 2010.[339] H.W. Gould. Bibliography on Euler’s constant. Unpublished manuscript.

[340] H.W. Gould. Some formulas for Euler’s constant. Bulletin of number theory 10(2):2–9, 1986.[341] P.J. Grabner, R. Tichy, U.T. Zimmermann. Inequalities for the Gamma function with ap-

plications to permanents. Discr. Math. 154:53–62, 1996.

[342] R.L. Graham, D.E. Knuth, O. Patashnik. Concrete mathematics. Addison-Wesley, ReadingMA, 1989.

[343] John Greene, Dennis Stanton. The triplication formula for Gauss sums. Aequationes Math.

[Aequationes Mathematicae] 30(2–3):134–141, 1986.[344] John V. Grice, Lee J. Bain. Inferences concerning the mean of the Gamma distribution. J.

Amer. Statist. Assoc. [Journal of the American Statistical Association] 75(372):929–933,

1980.[345] D. Gronau, J. Matkowski. Geometrically convex solutions of certain difference equations

and generalized Bohr-Mollerup theorems. Results Math. 26(3–4):290–297, 1994.

[346] Detlef Gronau, Janusz Matkowski. Geometrical convexity and generalizations of the Bohr-Mollerup theorem on the Gamma function. Math. Pannon. [Mathematica Pannonica]

4(2):153–160, 1993.

[347] Detlef Gronau, Janusz Matkowski. Geometrically convex solutions of certain difference equa-tions and generalized Bohr-Mollerup type theorems. Results in Mathematics 26:290–297,

1994.[348] T.H. Gronwall. On the convergence of Binet’s factorial series for log γ(s) and ψ(s). Annals

of Math. 18:74–78, 1916/17.

[349] T.H. Gronwall. The Gamma function in the integral calculus. Annals of Math. 20:35–124,1918/19.

[350] C. C. Grosjean. Formulae concerning the computation of the Clausen integral Cl2(θ). J.

Comput. Appl. Math. [Journal of Computational and Applied Mathematics] 11(3):331–342,1984.

[351] Benedict H. Gross, Neal Koblitz. Gauss sums and the p-adic γ-function. Ann. of Math. (2)

[Annals of Mathematics. Second Series] 109(3):569–581, 1979.[352] Johann Philipp Gruson. Supplement zu L. Eulers Differentialrechnung. [Supplement to L.

Euler’s Differential Calculus]. LTR Verlag GmbH viii+375 pp, DM 340.00 the four volume

set., Bad Honnef, 1981. Reprint of the 1798 edition.[353] Bandana Guha. Expansion of incomplete Gamma function in terms of Jacobi polynomials.

Indian J. Pure Appl. Math. [Indian Journal of Pure and Applied Mathematics] 12(8):990–993, 1981.

[354] B. N. Guo, F. Qi, H. M. Srivastava. Some uniqueness results for the non-trivially complete

montonicity of a class of functions involving the Polygamma and related functions. IntegralTransforms Spec. Funct. 21, 2010. Submitted by. H. M. Srivastava, 2. May 2010, (to appear).

[355] B.N. Guo, F. Qi. Inequalities and monotonicity for the ratio of Gamma functions. Taiwanese

J. Math. 7(2):239–247, 2003.[356] S. Guo. Monotonicity and concavity properties of some functions involving the Gamma

functions with applications. J. Ineq. Pure Appl. Math. 7(1), 2006.[357] S. Guo, F. Qi, H. M. Srivastava. Supplements to a class of logarithmically completely mono-

tonic functions associated with the Gamma function. Appl. Math. Comput. 197:768–774,

2008. Submitted by. H. M. Srivastava, 2. May 2010.

[358] A. Gupta. A q-extension of incomplete Gamma function II. Some further properties andassociated functions. J. Math. Phys. Sci. 28(6):271–285, 1994.

[359] Anju Gupta. A q-extension of “incomplete” Gamma function. J. Indian Math. Soc. (N.S.)[The Journal of the Indian Mathematical Society. New Series] 58(1–4):249–262, 1992.

[360] Arjun K. Gupta. On the expansion of bivariate Gamma distribution. J. Indian Statist.

Assoc. [Journal of the Indian Statistical Association] 17:41–50, 1979. Professor Huzurbazarfelicitation volume.


[361] Arjun K. Gupta. Some identities in γ-functions. Anal. Numer. Theor. Approx. [L’AnalyseNumerique et la Theorie de L’Approximation] 8(1):43–48, 1979.

[362] Arjun K. Gupta, C. F. Wong. On a Morgenstern-type bivariate Gamma distribution. Metrika

31(6):327–332, 1984.[363] C. P. Gupta, J. Mawhin. Asymptotic conditions at the two first eigenvalues for the periodic

solutions of Lienard differential equations and an inequality of E. Sch. Z. Anal. Anwendungen

[Zeitschrift fur Analysis und Ihre Anwendungen] 3(1):33–42, 1984.[364] H. L. Gupta. An integral equation involving incomplete Gamma function as the kernel.

Indian J. Pure Appl. Math. [Indian Journal of Pure and Applied Mathematics] 7(12):1361–1366, 1976.

[365] J. Gurland. An inequality satisfied by the Gamma function. Skand. Aktuarietidiskr 39:171–

172, 1956.[366] Andreas Guthmann. Asymptotische Entwicklungen fur Unvollstandige Gammafunktionen.

[asymptotic expansions for incomplete Gamma functions]. Forum Math. [Forum Mathe-

maticum] 3(2):105–141, 1991. German.[367] L. Habsieger. Une q-conjecture d’Askey. C.R. Acad. Sci. Paris 302:615–618, 1986.

[368] L. Habsieger. Conjecture de MacDonald et q-integrale de Selberg-Askey. PhD thesis, Univ.

Louis Pasteur, Strasbourg, 1987.[369] R. R. Hall, I. Vincze. On a simultaneous characterization of the Poisson law and the Gamma

distribution. In Collection: Analytical methods in probability theory (Oberwolfach, 1980),

volume 861 of Lecture Notes in Math, 54–59, Berlin–New York, 1981. Springer.[370] B. J. Harris. A note on the asymptotic series associated with a certain Titchmarsh-Weyl

m-function. Resultate Math. [Resultate der Mathematik. Results in Mathematics] 20(1–

2):499–506, 1991.[371] Gisela Hartler. Statistische Methode fur die Zuverlassigkeitsanalyse. [Statistical method for

the analysis of reliability]. VEB Verlag Technik, Berlin; distributed by Springer-Verlag,Vienna, 230 pp, 1983. In German.

[372] Gisela Hartler. The logarithmic Gamma distribution— A useful tool in reliability statistics.

In Collection: Transactions of the Tenth Prague Conference on Information Theory, Sta-tistical Decision Functions, Random Processes, Vol. A (Prague, 1986), 367–373, Dordrecht,

1988. Reidel.

[373] H. Haruki. A new characterization of Euler’s Gamma function by a functional equation.Aequationes Math. [Aequationes Mathematicae] 31(2–3):173–183, 1986.

[374] M. Hassani. An equivalent for prime number theorem. RGMIA Research Report Collection

8(2), 2005.[375] M. Hassani. Approximation of π(x) by ψ(x). J. Ineq. Pure Appl. Math. 7(1), 2006.

[376] Kazuyuki Hatada. On the values at rational integers of the p-adic Dirichlet l-functions. J.

Math. Soc. Japan [Journal of the Mathematical Society of Japan] 31(1):7–27, 1979.[377] J. Havil. Gamma. Exploring Euler’s constant. Princeton Univ. Press, 2003.[378] D. M. Hawkins. Identification of outliers. Chapman & Hall, x+188 pp, $24.95, London-New

York, 1980. Monographs on Applied Probability and Statistics.

[379] John H. Hawkins, Marvin I. Knopp. A Hecke-Weil correspondence theorem for automorphic

integrals on γ0(n), with arbitrary rational period functions. In Collection: A tribute to EmilGrosswald: number theory and related analysis, volume 43 of Contemp. Math, 451–475.

Amer. Math. Soc, Providence, RI, 1993.

[380] W.K. Hayman. A generalization of Stirling’s formula. J. Reine Angew. Math. 196:67–95,1956.

[381] Laxman M. Hegde, Ram C. Dahiya. Estimation of the parameters of a truncated Gammadistribution. Comm. Statist. Theory Methods [Communications in Statistics. Theory andMethods] 18(2):561–577, 1989.

[382] A. Helversen-Pasotto. Gamma-function and Gaussian-sum-function. Rend. Circ. Mat.

Palermo (2) Suppl. [Rendiconti del Circolo Matematico di Palermo. Serie II.Supplemento](30):195–200, 1993. Proceedings of the Winter School “Geometry and Physics”

(Srni, 1991).[383] Anna Helversen-Pasotto. L’identite de Barnes pour les corps finis. In Collection: Seminaire

Delange-Pisot-Poitou, 19e annee: 1977/78, Theorie des nombres, Fasc. 1, Exp. No. 22, 12

pp., Paris, 1978. Secretariat Math. French.


[384] Anna Helversen-Pasotto, Patrick Sole. Barnes’ first lemma and its finite analogue. Canad.Math. Bull. [Canadian Mathematical Bulletin. Bulletin Canadien de Mathematiques]

36(3):273–282, 1993.

[385] Guy Henniart. Cyclotomie et valeurs de la fonction γ (d’apres G. Anderson). [Cyclotomyand values of the γ-function (following G. Anderson)]. Asterisque 3(161–162):53–72, 1988.

Seminaire Bourbaki, 1987/88, Exp. No. 688, French.

[386] Ch. Hermite. Sur le logarithme de la fonction Gamma. American Journal Bd. 17:111–116,1895.

[387] Michel Herve. Les Fonctions Analytiques. [Analytic Functions]. Mathematiques. [Mathe-matics]. Presses Universitaires de France, 302 pp, 135 F., Paris, 1982. French.

[388] T.P. Hessami, Kh.P. Hessami. Criteria for irrationality of generalized Euler’s constant. J.

Number Theory 108(1):169–185, 2004.[389] Konrad J. Heuvers, Daniel Moak. Matrix solutions of the functional equation of the Gamma

function. Aequationes Math. [Aequationes Mathematicae] 33(1):1–17, 1987.

[390] Toyokazu Hiramatsu. On some dimension formula for automorphic forms of weight one. II.Nagoya-Math. J. [Nagoya Mathematical Journal] 105:169–186, 1987. With an addendum

to: “The modular equation and modular forms of weight one” [Nagoya Math. J. 100 (1985),

145–162] by Hiramatsu and Y. Mimura.[391] Wayne J. Holman III, Kenneth I. Gross. Matrix-valued special functions and representation

theory of the conformal group. I. the generalized Gamma function. Trans. Amer. Math. Soc.

[Transactions of the American Mathematical Society] 258(2):319–350, 1980.[392] Z.H. Hou. The total mean curvature of submanifolds in an Euclidean space. Michigan Math.

J. 45:497–505, 1998.

[393] Jian Huang. On some properties of functionals. Hunan Daxue Xuebao [Hunan Daxue Xue-bao. Ziran Kexue Ban. Journal of Hunan University. Natural Sciences] 20(1):12–17, 1993.

Chinese.[394] Wen Jang Huang, Li Sue Chen. Note on a characterization of Gamma distributions. Statist.

Probab. Lett. [Statistics and Probability Letters] 8(5):485–487, 1989.

[395] T. P. Hutchinson. Compound Gamma bivariate distributions. Metrika 28(4):263–271, 1981.[396] Tsvetan G. Ignatov. Properties of a class of point processes obtained from Euler’s formula.

In Collection: Mathematics and mathematical education (Sunny Beach, 1984), 335–342.

Blgar. Akad. Nauk, Sofia, 1984. Bulgarian.[397] Hideo Imai. Values of p-adic l-functions at positive integers and p-adic log multiple

Gamma functions. Tohoku Math. J. (2) [The Tohoku Mathematical Journal. Second Se-

ries] 45(4):505–510, 1993.[398] Christopher Olutunde Imoru. A note on an inequality for the Gamma function. Internat.

J. Math. Math. Sci. [International Journal of Mathematics and Mathematical Sciences]

1(2):227–233, 1978.[399] Makoto Ishibashi, Hisayoshi Sato, Katsumi Shiratani. On the Hasse invariants of elliptic

curves. Kyushu J. Math. [Kyushu Journal of Mathematics] 48(2):307–321, 1994.[400] M.E.H. Ismail. A remark on a paper by giordano, laforgia and pecaric. J. Math. Anal. Appl.

211(2):621–625, 1997.

[401] Mourad E. H. Ismail, Douglas H. Kelker. Special functions, Stieltjes transforms and infinitedivisibility. SIAM J. Math. Anal. [SIAM Journal on Mathematical Analysis] 10(5):884–901,

1979.

[402] Mourad E. H. Ismail, Lee Lorch, Martin E. Muldoon. Completely monotonic functionsassociated with the Gamma function and its q-analogues. J. Math. Anal. Appl. [Journal of

Mathematical Analysis and Applications] 116(1):1–9, 1986.[403] Mourad E. H. Ismail, Martin E. Muldoon. Inequalities and monotonicity proper-

ties for Gamma and q-Gamma functions. In Approximation and computation (West

Lafayette,IN,1993), volume 119 of Internat. Ser. Numer. Math, 309–323. Birkhauser,

Boston, 1994.[404] A. Ivic. The Riemann Zeta-function. John Wiley, 1985.

[405] A. Ivic, Z. Mijajlovic. On Kurepa’s problems in number theory. Publ. Inst. Math 57(71):19–28, 1995.

[406] F.H. Jackson. A generalization of the functions γ(n) and xn. Proc. Roy. Soc. London 74:64–

72, 1904.[407] F.H. Jackson. On q-definite integrals. Quart. J. Pure and Appl. Math. 41:193–203, 1910.


[408] F.H. Jackson. Certain q-identities. Quart. J. Math. 12:167–172, 1941.[409] L. Jacobsen. General convergence of continued fractions. Trans. Amer. Math. Soc. 294:477–

485, 1986.

[410] Lisa Jacobsen, W. B. Jones, H. Waadeland. Convergence acceleration for continued frac-tions k(an/1), where an → ∞. In Collection: Rational approximation and applications in

mathematics and physics (Lancut, 1985), volume 1237 of Lecture Notes in Math, 177–187,

Berlin-New York, 1987. Springer.[411] Herve Jacquet. Principal l-functions of the linear group. In Collection: Automorphic forms,

representations and L-functions (Proc. Sympos. Pure Math, Oregon State Univ, Corvallis,Ore., 1977), volume XXXIII of Proc. Sympos. Pure Math, Providence, R.I., 1979. Amer.

Math. Soc.

[412] V. K. Jain, H. M. Srivastava. Some families of multilinear q-generating functions and com-binatorial q-series identities. J. Math. Anal. Appl. 192:413–418, 1995. Submitted by. H. M.


[413] W. Janous. Inequalities for parts of the harmonic series. Univ. Beograd Publ. Elek. Fak. Ser.Mat. 12:38–40, 2001.

[414] J. L. Jensen. Inference for the mean of a Gamma distribution with unknown shape pa-

rameter. Scand. J. Statist. [Scandinavian Journal of Statistics. Theory and Applications]13(2):135–151, 1986.

[415] J. L. Jensen, L. B. Kristensen. Saddlepoint approximations to exact tests and improved

likelihood ratio tests for the Gamma distribution. Comm. Statist. Theory Methods [Com-munications in Statistics. Theory and Methods] 20(4):1515–1532, 1991.

[416] J.L.W.V. Jensen. An elementary exposition of the theory of the Gamma-function. Annals

of Math. (2) 17:124–166, 1916.[417] F. John. Special solutions of certain difference equations. Acta Math. 71:175–189, 1939.

[418] Frank Jones. Lebesgue Integration on Euclidean space. Jones and Bartlett Publishers,xvi+588 pp, $45.00., Boston, MA, 1993.

[419] W.B. Jones, W.J. Thron. On the computation of incomplete Gamma functions in the com-

plex domain. J. Comput. Appl. Math. 12–13:401–417, 1985.[420] W.B. Jones, W.J. Thron. Continued fractions in numerical analysis. Appl. Numerical Math.

4:143–230, 1988.

[421] F. Jongmans. Quelques pieces choisies dans la correspondance d’Eugene Catalan. [A selectionfrom the correspondence of Eugene Catalan]. Bull. Soc. Roy. Sci. Liege [Bulletin de la

Societe Royale des Sciences de Liege] 50(9–10):287–309, 1981. Correspondence: Eugene

Catalan, French.[422] Ch. Jordan. Calculus of finite differences. Budapest, 1939.

[423] Bent Jorgensen. Statistical properties of the generalized inverse Gaussian distribution., vol-

ume 4 of Memoirs. Aarhus Universitet, Matematisk Institut, 191 pp. (not consecutivelypaged), Aarhus, 1980.

[424] P. C. Joshi. On the moments of Gamma order statistics. Naval Res. Logist. Quart. [NavalResearch Logistics Quarterly] 26(4):675–679, 1979.

[425] Rajan L. Joshi, Thomas R. Fischer. Comparision of generalized Gaussian and Laplacian

modeling in DCT image coding. IEEE Signal Processing Letters 2(5):81–82, 1995.[426] Jean-Paul Jurzak. A class of generalized Gamma functions. Letters in Mathematical Physics

71:159–171, 2005.

[427] H. H. Kairies. Remarks concerning extensions of the Gamma function. In Collection: Gen-eral inequalities, 2 (Proc. Second Internat. Conf., Oberwolfach, 1978), 455–457, Basel-

Boston, Mass, 1980. Birkhauser.[428] Hans Heinrich Kairies. Zur Axiomatischen Charakterisierung der Gammafunktion. J. Reigne

Angew. Math 236:103–111, 1969. German.

[429] Hans Heinrich Kairies. Uber die Logarithmische Ableitung der Gammafunktion. Math. Ann

184:157–162, 1970. German.[430] Hans Heinrich Kairies. Definitionen der Bernoulli-Polynome mit Hilfe ihrer Multiplikation-

stheoreme. Manuscripta Math 8:363–369, 1973.[431] Hans Heinrich Kairies. Convexity in the Theory of the Gamma Function. In General In-

equalities (Proc. First Internat. Conf. Oberwolfach 1976), volume 1, 455–457. Birkhauser,

Basel-Boston, Mass, 1980.


[432] Hans Heinrich Kairies. Die Gammafunktion als stetige Losung eines Systems von Gauss-Funktionalgleichungen. [The Gamma function as continuous solution of a system of Gauss

functional equations]. Results Math. [Results in Mathematics. Resultate der Mathematik]

26(3–4):306–315, 1994. German.[433] H.H. Kairies. Charakterisierungen und Ungleichungen fur die q-Factorial-Funktionen. In

General Inequalities 4 (ISNM, vol.71), 257–267. Birkhauser, Basel, 1984.

[434] A. V. Kakosyan. Stability of a characterization of a generalized Gamma-distribution. InCollection: Stability problems for stochastic models, 58–61, Moscow, 1984. Vsesoyuz. Nauch.

Issled. Inst. Sistem. Issled. Russian.[435] R. N. Kalia. Fractional calculus: Its brief history and recent advances. In Collection: Recent

advances in fractional calculus, Global Res. Notes Ser. Math, 1–30. Global, Sauk Rapids,

MN, 1993.[436] R. N. Kalia, Sneh Kalia. The generalized Leibniz rule and its consequences. Panamer. Math.

J. [Panamerican Mathematical Journal] 1(2):97–99, 1991.

[437] Shyam L. Kalla. Integrals of generalized Jacobi functions. Proc. Nat. Acad. Sci. India Sect.A [Proceedings of the National Academy of Sciences, India. Section A] 58(1):123–128, 1988.

[438] M. Kaluszka. Admissible and minimax estimators of λr in the Gamma distribution with

truncated parameter space. Metrika 33(6):363–375, 1986.[439] V. N. Kalyuzhnyi. The power moment problem on a p-adic disk. Teor. Funktsii Funktsional.

Anal. i Prilozhen. [Kharkovskii Ordena Trudovogo Krasnogo Znameni Gosudarstvennyi

Universitet imeni A. M. Gorko], 1983. Russian.[440] Masanobu Kaneko. A generalization of the Chowla-Selberg formula and the Zeta functions

of quadratic orders. Proc. Japan Acad. Ser. A Math. Sci. [Japan Academy. Proceedings.

Series A. Mathematical Sciences] 66(7):201–203, 1990.[441] Ryuzo Kanno. Estimation in mixture of two Gamma distributions with equal scale param-

eters. TRU Math. [TRU Mathematics] 17(1):55–64, 1981.[442] H.J. Kanold. Ein einfacher Beweis der Stirlingschen Formel. Elem. Math. (Basel) 24(5):109–

110, 1969.

[443] J.N. Kapur. Logarithmic convexity of the Beta and Gamma functions. J. Nat. Acad. Math.India 3:67–77, 1985.

[444] A. A. Karatsuba. Osnovy Analiticheskoi Teorii Chisel. [Principles of Analytic Number The-

ory]. “Nauka”, 240 pp, 2.10 r., Moscow, 2nd edition, 1983. Russian.[445] A.A. Karatsuba, S.M. Voronin. The Riemann Zeta-function. de Gruyter, 1992.

[446] E. A. Karatsuba. A new method for the fast calculation of transcendental functions. Us-

pekhi Mat. Nauk [Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Us-pekhi Matematicheskikh-Nauk] 46(2):246–247, 1991. Translation: Russian-Math.-Surveys

[Russian-Mathematical-Surveys] 46 (1991), no. 2, 246–247.

[447] E.A. Karatsuba. On the computation of the Euler constant γ. Numer. Algorithms 24:83–87,2000.

[448] E.A. Karatsuba. On the asymptotic representation of the Euler Gamma function by Ra-manujan. J. Comput. Appl. Math. 135:225–240, 2001.

[449] D.K. Kazarinoff. On Wallis’ formula. Edinburgh Math. Notes(40):19–21, 1956.

[450] Jerome P. Keating, Ronald E. Glaser, Norma S. Ketchum. Testing hypotheses about theshape parameter of a Gamma distribution. Technometrics. A Journal of Statistics for the

Physical, Chemical and Engineering Sciences 32(1):67–82, 1990.

[451] C. Stuart Kelley. Accurate approximations to the Polygamma functions. Quart. Appl. Math.[Quarterly of Applied Mathematics] 37(2):203–207, 1979–1980.

[452] C. Stuart Kelley. Accurate approximations to the Polygamma functions. Quart. Appl. Math.[Quarterly of Applied Mathematics] 37(2):203–207, 1979–1980.

[453] A. W. Kemp. On Gamma function inequalities. Skand. Aktuarietidiskr:65–69, 1973.

[454] Rainer Kemp. Fundamentals of the Average Case Analysis of Particular Algorithms. Wiley-

Teubner Series in Computer Science. John Wiley & Sons, Ltd. and B. G. Teubner viii+233pp, $34.95., Chichester and Stuttgart, 1984.

[455] F.K. Kenter. A matrix representation for Euler’s constant γ. Amer. Math. Monthly 106:452–454, 1999.

[456] D. Kershaw. Some extensions of W. Gautschi’s inequalities for the Gamma function. Math.

Comp. [Mathematics of Computation] 41(164):607–611, 1983.


[457] Gotz Kersting. Asymptotic γ-distribution for stochastic difference equations. Stochastic Pro-cess. Appl. [Stochastic Processes and their Applications] 40(1):15–28, 1992.

[458] Jovan D. Keckic. A few remarks on automorphic functions. Publ. Inst. Math 30(44):69–71,

1981.[459] Jovan D. Keckic, Miomir S. Stankovic. Some inequalities for special functions. Publ. Inst.

Math 13(27):51–54, 1972.

[460] Abdul Hamid Khan, R. U. Khan. Recurrence relations between moments of order statis-tics from generalized Gamma distribution. J. Statist. Res. [Journal of Statistical Research]

17(1–2):75–82, 1983.[461] M. S. H. Khan. A generalized Gamma distribution and some of its properties. Libyan J.

Sci. [The Libyan Journal of Science] 12:53–60, 1982–1983.

[462] C. G. Khatri. Characterization of Gamma and g-binomial distributions based on regressions.J. Indian Statist. Assoc. [Journal of the Indian Statistical Association] 22:115–128, 1984.

[463] A. Yu. Khrennikov. Analysis on the p-adic superspace. I. Generalized functions: Gaussian

distribution. J. Math. Phys. [Journal of Mathematical Physics] 33(5):1636–1642, 1992.[464] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo. Theory and Applications of Fractional Dif-

ferential Equations, volume 204 of North-Holland Mathematical Studies. Elsevier (North-

Holland) Science Publishers, Amsterdam, London and New York, 2006. Submitted by. H.M. Srivastava, 2. May 2010.

[465] Keeyoung Kim. Probability integral of the inverted Dirichlet distribution with application.

J. Korean Statist. Soc. [Journal of the Korean Statistical Society] 13(1):25–31, 1984.[466] T. Kim. On a q-analogue of the p-adic log Gamma functions and related integrals. J. Number

Theory 76(2):320–329, 1999.

[467] T. Kim. q-Volkenborn integration. Russ. J. Math. Phys. 9:288–299, 2002.[468] T. Kim, C. Adiga. On the q-analogue of Gamma functions and related inequalities. J. Ineq.

Pure Appl. Math. 6(4), 2005.[469] Y.S. Kim. q-analogues of p-adic Gamma functions and p-adic Euler constants. Commun.

Korean Math. Soc. 13(4):735–741, 1998.

[470] V.H. Kinkelin. Uber eine mit der Gamma Funktion verwandte Transcendente und derenAnwendungen auf die Integralrechnung. J.Reine Angew.Math.:122–158, 1860.

[471] M.S. Klamkin. Extensions of Dirichlet’s multiple integral. SIAM J. Math. Anal. 2:467–469,

1971.[472] Fima C. Klebaner. Stochastic difference equations and generalized Gamma distributions.

Ann. Probab. [The Annals of Probability] 17(1):178–188, 1989.[473] Heinz Jurgen Klemmt. Ein Masstheoretischer Satz mit Anwendungen auf Funktionalgle-

ichungen. [A measure-theoretic theorem with applications to functional equation]. Arch.

Math. (Basel) [Archiv der Mathematik. Archives of Mathematics. Archives Mathematiques]36(6):537–540, 1981. German.

[474] M. Kline. Mathematical thought from ancient to modern times. Oxford Univ. Press, 1972.[475] M.I. Knopp. Modular functions in analytic number theory. Markham, Chicago, 1970.[476] Kazuya Kobayashi. On generalized Gamma functions occurring in diffraction theory. J.

Phys. Soc. Japan [Journal of the Physical Society of Japan] 60(5):1501–1512, 1991.

[477] N. Koblitz. Interpretation of the p-adic log Gamma function and Euler constants using theBernoulli measure. Trans. Amer. Math. Soc. 242:261–269, 1978.

[478] N. Koblitz. P-adic analysis: A Short Course on Recent Work, volume 46 of London Math.

Soc. Lecture Notes Series. London Math. Society, 1980.[479] N. Koblitz. q-extension of the p-adic Gamma function. Trans. Amer. Math. Soc. 260:449–

457, 1980.[480] Neal Koblitz. A new proof of certain formulas for p-adic l-functions. Duke Math. J. [Duke

Mathematical Journal] 46(2):455–468, 1979.

[481] Neal Koblitz. The hypergeometric function with p-adic parameters. In Collection: Pro-ceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont, 1979), Queen’s

Papers in Pure and Appl. Math, 54, 319–328, Kingston, Ont, 1980. Queen’s Univ, 1980.

[482] Neal Koblitz. P -adic analog of Heine’s hypergeometric q-Series. Pacific J. Math. [PacificJournal of Mathematics] 102(2):373–383, 1982.

[483] Neal Koblitz. Q-extension of the p-adic Gamma function. II. Trans. Amer. Math. Soc.

[Transactions of the American Mathematical Society] 273(1):111–129, 1982.


[484] Neal Koblitz. P -adic Numbers, p-adic Analysis, and Zeta-functions., volume 58 of GraduateTexts in Mathematics. Springer-Verlag, xii+150 pp, $24.80., New York-Berlin, 2nd edition,

1984.

[485] V.L. Kocic. A note on q-Gamma function. Univ. Beograd Publ. Elektr. Fak. Ser. Mat. 1:31–34, 1990.

[486] Vlajko Lj. Kocic. A note on q-Gamma function. Univ. Beograd. Publ. Elektrotehn. Fak. Ser.

Mat. [Univerzitet u Beogradu. Publikacije Elektrotehnickog Fakulteta. Serija Matematika]1, 1990.

[487] V. A. Kofanov. Approximation by algebraic polynomials of classes of functions that arefractional integrals of summable functions. Anal. Math. [Analysis Mathematica] 13(3):211–

229, 1987. Russian.

[488] V. A. Kofanov. Approximation by algebraic polynomials of classes of functions that arefractional integrals of summable functions. Anal. Math. [Analysis Mathematica] 13(3):211–

229, 1987. Russian.

[489] Maia Koicheva. A characterization of the Gamma distribution in terms of conditional mo-ment. Appl. Math. [Applications of Mathematics] 38(1):19–22, 1993.

[490] Maia Koicheva. On an inequality and the related characterization of the Gamma distribution.

Appl. Math. [Applications of Mathematics] 38(1):11–18, 1993.[491] Tom H. Koornwinder. Jacobi functions as limit cases of q-ultraspherical polynomials. J.

Math. Anal. Appl. [Journal of Mathematical Analysis and Applications] 148(1):44–54, 1990.

[492] Adam Koranyi. The volume of symmetric domains, the Koecher Gamma function and anintegral of Selberg. Studia Sci. Math. Hungar 1, 1982.

[493] Bertram Kostant, Siddhartha Sahi. The Capelli identity, Tube domains, and the generalized

Laplace transform. Adv. Math. [Advances in Mathematics] 87(1):71–92, 1991.[494] Stamatis Koumandos. Remarks on a paper by Chao-Ping Chen and Feng Qi. Proceedings

of the American Mathematical Society 134(5):1365–1367, 2005. Submitted by. StamatisKoumandos, 27. April 2010, Article electronically published on October 6, 2005, S 0002-

9939(05)08104-9.

[495] Stamatis Koumandos. Remarks on some completely monotonic functions. J. Math. Anal.Appl. 324:1458–1461, 2006. Submitted by. Stamatis Koumandos, 27. April 2010, Available

online 4 January 2006.

[496] Stamatis Koumandos. Monotonicity of some functions involving the Gamma and Psi func-tions. Mathematics of Computation S 0025-5718(08)02140-6 77(264):2261–2275, October

2008. Submitted by. Stamatis Koumandos, 27. April 2010, Article electronically published

on May 14, 2008.[497] Stamatis Koumandos. On Ruijsenaars’ asymptotic expansion of the logarithm of the dou-

ble Gamma function. J. Math. Anal. Appl. 341:1125–1132, 2008. Submitted by. Stamatis

Koumandos, 27. April 2010, Available online 17 November 2007.[498] Stamatis Koumandos, Martin Lamprecht. Some completely monotonic functions of posi-

tive order. Mathematics of Computation S 0025-5718 (09) 02313-8, 2009. Submitted by.Stamatis Koumandos, 27. April 2010, Article electronically published on November 9, 2009.

[499] Stamatis Koumandos, Henrik L. Pedersen. Completely monotonic functions of positive or-

der and asymptotic expansions of the logarithm of Barnes double Gamma function andEuler’s Gamma function. J. Math. Anal. Appl. 355:33–40, 2009. Submitted by. Stamatis

Koumandos, 27. April 2010, Available online 5. February 2009.

[500] Stamatis Koumandos, Henrik L. Pedersen. On the asymptotic expansion of the logarithmof Barnes triple Gamma function. Math. Scand. 105:287–306, 2009. Submitted by. Stamatis

Koumandos, 27. April 2010.[501] M. A. Kovalevskii. The asymptotic behavior of functions that generalize the Euler Gamma-

function. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) [Zapiski Nauch-

nykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institut Steklov.], 1981.

Mathematical questions in the theory of wave propagation, 11., In Russian.[502] M. A. Kovalevskii. Construction of asymptotic expressions for some entire functions. Vest-

nik Leningrad. Univ. Mat. Mekh. Astronom. [Vestnik Leningradskogo Universiteta. Matem-atika, Mekhanika, Astronomiya] 1(1), 1985.

[503] Teresa Kowalczyk, Joanna Tyrcha. Multivariate Gamma distributions—properties and

shape estimation. Statistics. A Journal of Theoretical and Applied Statistics 20(3):465–474, 1989.


[504] K. H. Kraft, E. Schnieder. Rationale Rechenoperationen mit Gammaverteilten Zufallsvari-ablen. Siemens Forsch. Entwickl. [Siemens Forschungs und Entwicklungsberichte. Research

and Development Reports] 9(4):227–230, 1980. German.

[505] Walter Stephan Kramer. Berechnung der Gammafunktion γ(x) fur reelle Punkt–und Inter-vallargumente. [Computation of the Gamma function γ(x) for real point and interval ar-

guments]. Z. Angew. Math. Mech. [Zeitschrift fur Angewandte Mathematik und Mechanik]

70(6):581–584, 1990. Bericht uber die Wissenschaftliche Jahrestagung der GAMM (Karl-sruhe,1989)., German.

[506] C. Krattenthaler, H. M. Srivastava. Summations for basic hypergeometric series involving aq-analogue of the Digamma function. Comput. Math. Appl. 32(3):73–91, 1996. Submitted

by. H. M. Srivastava, 2. May 2010.

[507] E. Kratzel. Lattice Points. WEB Deutscher Verlag der Wissenschaften, Berlin, 1988.[508] Wolfgang Krull. Bemerkungen zur Differenzengleichung g(x + 1) − g(x) = ϕ(x). Math.

Nachrichten 1:365–367, 1948. German.

[509] Daniel S. Kubert, Stephen Lichtenbaum. Jacobi-sum Hecke characters and Gauss-sum iden-tities. Compositio Math. [Compositio Mathematica] 48(1):55–87, 1983.

[510] V. I. Kucheryavyi. Simple α-parameter integrals of renormalized Feynman amplitudes. Dokl.

Akad. Nauk Ukrain. SSR Ser. A [Akademiya Nauk Ukrainskoi SSR. Doklady. Seriya A.Fiziko Matematicheskie i Tekhnicheskie Nauki], 1983. Russian.

[511] Marek Kuczma. Remarques sur les quelques theoremes de J. Anastassiadis. Bull. Sci. Math,

2e serie 84:98–102, 1960. French.[512] Marek Kuczma. Functional Equations in a Single Variable. Polish Scientific Publishers,

Warszawa, 1968.

[513] P. Kumar, S.P. Singh, S.S. Dragomir. Some inequalities involving Beta and Gamma func-tions. Nonlinear Analysis Forum 6(1):143–150, 2001.

[514] E.E. Kummer. Beitrag zur Theorie der Funktion γ(x). J. Reine Angew. Math. 35:1–4, 1947.[515] A. I. Kurdjumova. The Bolzano-Cauchy criterion for convergence in an 1812 work of Gauss.

Istor. Mat. Issled. [Istoriko Matematicheskie Issledovaniya](23):142–143, 358, 1978. Rus-

sian.[516] D- . Kurepa. Left factorial function in the complex domain. Math. Balkanica 3:297–307, 1973.

[517] D- uro Kurepa. On the left factorial function !n. Mathematica Balcanica 1:147–153, 1971.

[518] A. F. Kurgaev, K. Zh. Tsatryan. Realizatsiya i ispolzovanie Γ-funktsii dlya vychislenii shi-rokogo nabora spetsialnykh funktsii. Preprint, 84–66. Akad. Nauk Ukrain. SSR, Inst. Kiber-

net. 24 pp, Kiev, 1984.

[519] Nobushige Kurokawa. Parabolic components of Zeta functions. Proc. Japan Acad. Ser. AMath. Sci. [Japan Academy. Proceedings. Series A. Mathematical Sciences] 64(1):21–24,

1988.

[520] Nobushige Kurokawa. Multiple Zeta functions: An example. In Collection: Zeta functions ingeometry (Tokyo, 1990), volume 21 of Adv. Stud. Pure Math, 219–226. Kinokuniya, Tokyo,1992.

[521] P. I. Kuznetsov, A. S. Yudina. Asymptotic expansions of solutions of a degenerate hypergeo-

metric equation. Differentsialnye Uravneniya [Differentsialnye Uravneniya] 23(3):428–434,

547., 1987. Russian.

[522] Gilbert Labelle. A propos d’un q-analogue pour la fonction Gamma d’Euler. [A q-analogue

for the Euler Gamma function]. Ann. Sci. Math. Quebec [Les Annales des Sciences Mathe-

matiques du Quebec] 6(2):163–196, 1982. French.[523] Andrea Laforgia. Further inequalities for the Gamma function. Math. Comp. [Mathematics

of Computation] 42(166):597–600, 1984.[524] Andrea Laforgia, Silvana Sismondi. A geometric mean inequality for the Gamma function.

Boll. Un. Mat. Ital. A (7) [Unione Matematica Italiana. A. Serie VII] 3(3):339–342, 1989.

[525] C. D. Lai, Terry Moore. Probability integrals of a bivariate Gamma distribution. J. Statist.Comput. Simulation [Journal of Statistical Computation and Simulation] 19(3):205–213,

1984.

[526] K. Lajko. A characterization of generalized normal and Gamma distributions. In Collection:Analytic function methods in probability theory (Proc. Colloq. Methods of Complex Anal. in

the Theory of Probab. and Statist., Kossuth L. Univ. Debrecen, Debrecen, 1977), volume 21

of Colloq. Math. Soc. Janos Bolyai, 199–225, Amsterdam–New York, 1979. North–Holland.


[527] C. Lanczos. A precision approximation of the Gamma function. SIAM J. Numer. Anal.1:86–96, 1964.

[528] E. Landau. Zur Theorie der Gammafunktion. Journal fur Math. 123:276–283, 1901.

[529] Serge Lang. Cyclotomic Fields. II., volume 69 of Graduate Texts in Mathematics. SpringerVerlag, xi+164 pp, $19.80., New York-Berlin, 1980.

[530] L. J. Lange. Continued fraction representations for functions related to the Gamma function.

In Collection: Continued fractions and orthogonal functions (Loen, 1992), volume 154 ofLecture Notes in Pure and Appl. Math., 233–279. Dekker, New York, 1994.

[531] Detlef Laugwitz, Bernd Rodewald. Auf Eulers Spuren zur Gammafunktion. [Following in Eu-ler’s tracks to the Gamma function]. Math. Semesterber. [Mathematische Semesterberichte]

33(2):201–212, 1986. German.

[532] Detlef Laugwitz, Bernd Rodewald. A simple characterization of the Gamma function. Amer.Math. Monthly [The American Mathematical Monthly] 94(6):534–536, 1987.

[533] J. L. Lavoie. Some evaluations for the generalized Hypergeometric Series. Math. Comp.

[Mathematics of Computation] 46(173):215–218, 1986.[534] M. Lavrentiev, B. Chebat. Methodes de la theorie des fonctions d’un variables complexe.

Editions Mir, Moscou, 1972.

[535] J. F. Lawless. Inference in the generalized Gamma and log Gamma distributions. Tech-nometrics. A Journal of Statistics for the Physical, Chemical and Engineering Sciences

22(3):409–419, 1980.

[536] I. Lazarevic, A. Lupas. An inequality for the Gamma function. Gaz. Mat. Perf. Met. 3(1–2):82–84, 1982.

[537] I. Lazarevic, M. Merkle. An Inequality Involving ERF and hyperbolic functions. Bulletin

Mathematique (Bucuresti) 36(84):317–318, 1992.[538] Ilija Lazarevic, Alexandru Lupas. Functional equations for Wallis and Gamma functions.

Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz 461-497:245–251, 1974.[539] D.H. Lehmer. Euler constants for arithmetical progressions. Acta Arith. 27:125–142, 1975.

[540] Lev J. Leifman, editor. Selected translations in mathematical statistics and probability. Vol.

15. American Mathematical Society, v+317 pp, $50.00., Providence, R.I., 1981.[541] W. Leighton. Remarks on certain Eulerian constants. Amer. Math. Monthly 75:283–285,

1968.

[542] Roy Leipnik. Subvex functions and Bohr’s Uniqueness Theorem. American MathematicalMonthly:1093–1094, 1967.

[543] M. Lerch. Theorie der Gammafunktion. Vestnik, 1899. 17 pp.

[544] R. T. Leslie, A. Khalique. A note on bias reduction in moment estimators of the Gammadistribution. Pakistan J. Statist. [Pakistan Journal of Statistics] 2(2):33–40, 1986.

[545] Gerard Letac. A characterization of the Gamma distribution. Adv. in Appl. Probab. [Ad-

vances in Applied Probability] 17(4):911–912, 1985.[546] L. Lewin. Polylogarithms and associated functions. North Holland, 1981.[547] Gang Li, Kai Tai Fang. The Ramanujan’s q-extension for the exponential function and statis-

tical distributions. Acta Math. Appl. Sinica (English Ser.) [Acta Mathematicae Applicatae

Sinica. English Series. Yingyong Shuxue Xuebao] 8(3):264–280, 1992.

[548] Wen Ch’ing Winnie Li. Barnes’ identities and representations of gl(2). II. Non-Archimedeanlocal field case. J. Reine Angew. Math. [Journal fur die Reine und Angewandte Mathematik]

345:69–92, 1983.

[549] Stephen Lichtenbaum. Values of l-functions of Jacobi-sum Hecke characters of Abelian fields.In Collection: Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981),

Progr. Math 26, 207–218. Birkhauser, Boston, MA, 1982.[550] N. A. Likhoded, P. I. Sobolevskii. Composite formulas for approximate calculation of con-

tinual integrals according to an infinite-dimensional γ-distribution. Vestsi Akad. Navuk

BSSR Ser. Fiz. Mat. Navuk [Vestsi Akademii Navuk BSSR. Seryya Fizika Matematych-

nykh Navuk] 5(5):37–43, 139., 1979. Russian.[551] H. Limbourg. Theorie de la fonction Gamma. Gent, 1859.

[552] Frank Lin, Zinne Jones. Numerical calculation of the Gamma function. Menemui Mat.[Menemui Matematik. Discovering Mathematics] 11(1):10–19, 1989.

[553] S. D. Lin, Y. S. Chao, H. M. Srivastava. Some expansions of the exponential integral in

series of the incomplete Gamma function. Appl. Math. Lett. 18:513–520, 2005. Submittedby. H. M. Srivastava, 2. May 2010.


[554] S. D. Lin, T. M. Hsieh, H. M. Srivastava. Some families of multiple infinite sums associatedwith the Digamma and related functions. J. Fract. Calc. 24:77–85, 2003. Submitted by. H.

M. Srivastava, 2. May 2010.

[555] S. D. Lin, H. M. Srivastava. Fractional calculus and its applications involving bilateralexpansions and multiple infinite series. J. Fract. Calc. 25:47–58, 2004. Submitted by. H. M.


[556] S. D. Lin, H. M. Srivastava. Some closed-form evaluations of multiple hypergeometric and q-hypergeometric series. Acta Appl. Math. 86:309–327, 2005. Submitted by. H. M. Srivastava,

2. May 2010.[557] S. D. Lin, H. M. Srivastava, P. Y. Wang. Some families of series and integrals associated

with the Digamma and Zeta functions. J. Fract. Calc. 23:121–131, 2003. Submitted by. H.

M. Srivastava, 2. May 2010.[558] S. D. Lin, H. M. Srivastava, P. Y. Wang. Some multiple infinite sums and combinatorial

identities derived by means of fractional calculus. J. Fract. Calc. 28:75–86, 2005. Submitted

by. H. M. Srivastava, 2. May 2010.[559] S. D. Lin, S. T. Tu, H. M. Srivastava. Certain classes of infinite sums evaluated by means

of fractional calculus operators. Taiwanese J. Math. 6:475–498, 2002. Submitted by. H. M.

Srivastava, 2. May 2010.[560] Ernst Lindelof. Le calcul des residus et ses applications a la theorie des fonctions. [The

Calculus of Residues and its Applications to the Theory of Functions]. Les Grands Clas-

siques Gauthier-Villars. [Gauthier-Villars Great Classics]. Editions Jacques Gabay, viii+145pp, 180 F, Sceaux, 1989. Reprint of the 1905 original., French.

[561] G. S. Lingappaiah. On the piecewise Gamma distribution. Tamkang J. Math. [Tamkang

Journal of Mathematics] 12(1):7–14, 1981.[562] G. S. Lingappaiah. Bivariate Gamma distribution as a life test model. Apl. Mat.

[Ceskoslovenska Akademie Ved. Aplikace Matematiky] 29(3):182–188, 1984.[563] G. S. Lingappaiah. Testing scale parameter in the Gamma distribution. Publ. Inst. Statist.

Univ. Paris [Publications de l’Institut de Statistique de l’Universite de Paris] 29(2):35–47,

1984.[564] G. S. Lingappaiah. Cyclical life tests with Gamma distribution as the test model. Demon-

stratio Math. [Demonstratio Mathematica] 18(2):421–435, 1985.

[565] J. Liouville. Note sur quelques integrales definie. J. Math. Pures Appl.,(1) 4:225–235, 1839.[566] G. Lochs. Supplement to the paper: “On the coefficients of the asymptotic series for the

correction factor of the Stirling formula and a special value of the incomplete Gamma func-

tion” , Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. ii 196 (1987), no. 1-3, 27–37.Osterreich. Akad. Wiss. Math. Natur. Kl. Sitzungsber. II [Osterreichische-Akademie-

der-Wissenschaften-Mathematische -Naturwissenschaftliche-Klasse.-Sitzungsberichte.-

Abteilung II. -Mathematische,-Physikalische-und-Technische-Wissenschaften] 196(8–10),1987. In German.

[567] Gustav Lochs. Uber die Koeffizienten der Asymptotischen Reihen fur den Korrekturfaktorder Stirlingschen Formel und einen speziellen Wert der unvollstandigen Gammafunktion.

[On the coefficients of the asymptotic series for the correction factor of the Stirling formula

and a special value of the incomplete Gamma function]. Osterreich. Akad. Wiss. Math.Natur. Kl. Sitzungsber. II [Osterreichische Akademie der Wissenschaften Mathematische

Naturwissenschaftliche Klasse. Sitzungsberichte. Abteilung II. Mathematische, Physikalis-

che und Technische Wissenschaften] 196(1–3):27–37, 1987. In German.[568] Francois Loeser. Arrangements d’hyperplans et sommes de Gauss. [Arrangements of hyper-

planes and Gauss sums]. Ann. Sci. Ecole Norm. Sup. (4) [Annales Scientifiques de l’EcoleNormale Superieure. Quatrieme Serie] 24(4):379–400, 1991. French.

[569] L. Lorch. Comparison of a pair of upper bounds for a ratio of Gamma functions. Math.

Balkanica (N.S.) 16:195–202, 2002. Fasc.1–4.[570] Lee Lorch. Inequalities for ultraspherical polynomials and the Gamma function. J. Approx.

Theory [Journal of Approximation Theory] 40(2):115–120, 1984.[571] L. Lorentzen, H Waadeland. Continued fractions with applications. North Holland, Amster-

dam, 1992.[572] F. Losch, F. Schoblik. Die Fakultat (Gammafunktion) und verwandte Funktionen. Leipzig,

1951.


[573] Lutz G. Lucht. Mittelwertungleichungen fur Losungen gewisser Differenzengleichungen.[Mean value inequalities for solutions of certain difference equations]. Aequationes Math.

[Aequationes Mathematicae] 39(2–3):204–209, 1990. German.

[574] Y.L. Luke. The special functions and their approximations. Academic Press, New York,1969.

[575] Y.L. Luke. Evaluation of the Gamma function by means of Pade approximations. SIAM J.

Math. Anal. 1:266–281, 1970.[576] Y.L. Luke. Inequalities for the Gamma function and its logarithmic derivative. Math. Balkan-

ica 2:118–123, 1972.[577] Yudell L. Luke. Mathematical Functions and their Approximations. Academic Press, New

York - San Francisco - London, 1975.

[578] Yudell L. Luke. On a summability method. In Collection: Pade approximation and itsapplications (Proc. Conf, Univ. Antwerp, 1979), volume 765 of Lecture Notes in Math,

295–337, 1979.

[579] Yudell L. Luke, Jet Wimp, Bing Y. Ting. Solution of difference equations by use of theτ -method. J. Approx. Theory [Journal of Approximation Theory] 32(3):211–225, 1981.

[580] A. Lupas. Mean value theorems for the Fourier-Jacobi coefficients (Romanian). Rev. Anal.

Numer. Theor. Approx. (Cluj) 3:79–84, 1974.[581] A. Lupas. Open Problems (Romanian). Gaz. Mat. (Bucuresti) 2, 1979.

[582] A. Lupas, L. Lupas. On certain special functions (Romanian). In Seminarul itinerant ec.

functionale, aprox., convexitate, 55–68, Timisoara, 1980.[583] A. Lupas, A. Vernescu. On an inequality (Romanian). Gaz. Mat. A 17(3):201–210, 1999.

[584] Luciana Lupas. The G-function of E. W. Barnes. Univ. Beograd. Publ. Elektrotehn. Fak. Ser.

Mat. [Univerzitet u Beogradu. Publikacije Elektrotehnickog Fakulteta. Serija Matematika]1:5–11, 1990.

[585] Lan Ma. γ-functions on a class of nonselfadjoint cones, and their applications. J. ChinaUniv. Sci. Tech. [Journal of China University of Science and Technology. Zhongguo Kexue

Jishu Daxue Xuebao] Math. Issue 13, 1983. In Chinese.

[586] Gordon Wilson MacDonald. Hilbert spaces of q-shifted factorial series. J. Math. Anal. Appl.[Journal of Mathematical Analysis and Applications] 178(1):30–56, 1993.

[587] W. Magnus et al. Formulas and theorems for the special functions of mathematical physics.

Springer Verlag, 3rd edition, 1966.[588] K. Mahler. On two analytic functions. Acta Arith. [Polska Akademia Nauk. Instytut Matem-

atyczny. Acta Arithmetica] 49(1):15–20, 1987.

[589] B.J. Malesevic. Some inequalities for Kurepa’s function. J. Ineq. Pure Appl. Math. 5(4),2004.

[590] P. Mansion. Sur la fonction Gamma. Annales de la Soc. Scient. Bruxelles 17:4–8, 1893.

[591] O. I. Marichev, Kim Tuan Vu. Some properties of the q-Gamma function γq(z). Dokl. Akad.Nauk BSSR [Doklady Akademii Nauk BSSR] 26(6):488–491, 571, 1982. Russian.

[592] O.I. Marichev, Vu Kim Tuan. Some properties of the q-Gamma function. Dokl. Akad. NaukBSSR 26(6):488–491, 1982.

[593] Clemens Markett, James Rovnyak. An incomplete Gamma transform and applications to

factorial series. Analysis [Analysis. International Mathematical Journal of Analysis and itsApplications] 12(1–2):159–186, 1992.

[594] George Marsaglia. The x+y, x/y characterization of the Gamma distribution. In Collection:

Contributions to probability and statistics, 91–98. Springer, New York-Berlin, 1989.[595] Albert W. Marshall, Ingram Olkin. Maximum likelihood characterizations of distributions.

Statist. Sinica [Statistica Sinica] 3(1):157–171, 1993.[596] M. S. Maswadah. Two approaches based on the structural model to inference on the general-

ized Gamma parameters. Egyptian Statist. J. [The Egyptian Statistical Journal] 33(1):110–

129, 1989.

[597] M. S. Maswadah. Structural inference on the generalized Gamma distribution based on type-II progressively censored sample. J. Austral. Math. Soc. Ser. A [Australian Mathematical

Society. Journal. Series A] 50(1):15–22, 1991.[598] A. M. Mathai. Special functions of matrix arguments and statistical distributions. Indian

J. Pure Appl. Math. [Indian Journal of Pure and Applied Mathematics] 22(11):887–903,

1991.


[599] A. M. Mathai, P. G. Moschopoulos. On a multivariate Gamma. J. Multivariate Anal. [Jour-nal of Multivariate Analysis. An International Journal] 39(1):135–153, 1991.

[600] A. M. Mathai, P. G. Moschopoulos. A form of multivariate Gamma distribution. Ann. Inst.

Statist. Math. [Annals of the Institute of Statistical Mathematics] 44(1):97–106, 1992.[601] K. Matsumoto. Asymptotic series for double Zeta and double Gamma functions of Barnes.

Analytic number theory (Japanese) (Kyoto, 1994)(958):162–165, 1996.

[602] K. Matsumoto. Asymptotic series for double Zeta, double Gamma, and Hecke L-functions.Math. Proc. Cambr. Phil. Soc. 123:385–405, 1998.

[603] Y. Matsuoka. Generalized Euler constants associated with the Riemann Zeta function. InNumber Theory and Combinatorics, Proc. 1984 Tokyo Conf., 279–295, 1984.

[604] D.S. McAnally. q-exponential functions and q-Gamma functions. In Confronting the infinite

(Adelaide, 1994), 246–253, River Edge, NJ, 1995. World Sci. Publ.[605] Adam C. McBride. On an index law and a result of Buschman. Proc. Roy. Soc. Edinburgh

Sect. A 96, 1984.

[606] J. H. McCabe. On an asymptotic series and corresponding continued fraction for a Gammafunction ratio. J. Comput. Appl. Math. [Journal of Computational and Applied Mathemat-

ics] 9(2):125–130, 1983.

[607] Peter McCullagh. A rapidly convergent series for computing ψ(z) and its derivatives. Math.Comp. [Mathematics of Computation] 36(153):247–248, 1981.

[608] A. McD. Mercer. Some new inequalities for the Gamma, Beta and Zeta functions. J. Ineq.

Pure Appl. Math. 7(1), 2006.[609] Ed McKenzie. Product autoregression: A time-series characterization of the Gamma distri-

bution. J. Appl. Probab. [Journal of Applied Probability] 19(2):463–468, 1982.

[610] G. Meinardus. An approximation theoretic alternative to asymptotic expansions for specialfunctions. Comput. Math. Appl. Part B [Computers and Mathematics with Applications.

An International Journal. Part B] 12(5–6):1191–1201, 1986.[611] H. Mellin. Zur Theorie der Gammafunktion. Acta Mathematica 8:37–80, 1886.

[612] V. P. Memnonov. A remark on estimates for Gamma functions and their ratios. Vestnik

Leningrad. Univ. Mat. Mekh. Astronom. [Vestnik Leningradskogo Universiteta. Matem-atika, Mekhanika, Astronomiya] 1:58–63, 118., 1988. In Russian.

[613] M. Mendes France, J.J. Jia. Transcendence and the Carlitz-Goss Gamma function. J. Num-

ber Theory 63(2):396–402, 1997.[614] Armel Mercier. Identities containing Gaussian binomial coefficients. Discrete Math. [Discrete

Mathematics] 76(1):67–73, 1989.

[615] M. Merkle. Some Inequalities for the Chi Squared Distribution Function. Univ. Beograd,Publ. Elektrotehn. Fak. Ser. Mat. 2:89–94, 1991.

[616] M. Merkle. Some inequalities for the Chi Square distribution function and the exponential

function. Arch. Math. 60:451–458, 1993.[617] M. Merkle. Logarithmic convexity and inequalities for the Gamma function. J. Math. Anal-

ysis Appl. 203:369–380, 1996.[618] M. Merkle. Inequalities for residuals of power expansions for the exponential function and

completely monotone functions. J. Math. Analysis Appl. 212:126–134, 1997.

[619] M. Merkle. Conditions for convexity of a derivative and some applications to the Gammafunction. Aequ. Math. 55:273–280, 1998.

[620] M. Merkle. Convexity, Schur-convexity and bounds for the Gamma function involving the

Digamma function. Rocky Mountain J. Math. 28(3):1053–1066, 1998.[621] M Merkle. Convolutions of logarithmically concave functions. Univ. Beograd, Publ. Elek-

trotehn. Fak. Ser. Mat. 9:113–117, 1998.[622] M. Merkle. Representation of the error term in Jensen’s and some related inequalities with

applications. J. Math. Analysis Appl. 231:76–90, 1999.

[623] M Merkle. Conditions for convexity of a derivative and applications to the Gamma and

Digamma function. Facta Universitatis (Nis), Ser. Math. Inform. 16:13–20, 2001.[624] M. Merkle. Reciprocally convex functions. J. Math. Analysis Appl. 293:210–218, 2004.

[625] M. Merkle. Gurland’s ratio for the Gamma function. Computers and Math. with Applications49:389–406, 2005.

[626] M. Merkle. Convexity in the theory of the Gamma function. International Journal of Applied

Mathematics and Statistics 11(7):103–117, 2007.


[627] M. Merkle. Inequalities for the Gamma function via convexity. In P. Cerone, S.S. Dragomir,editors, ”Advances in Inequalities for Special Functions”, 81–100. Nova Science Publishers,

Hauppauge-New York, 2008.

[628] M. Merkle, M. Moulin Ribeiro Merkle. Krull’s theory for the double Gamma function.Applied Mathematics and Computation 218:935–943, 2011. Submitted by. Milan Merkle,

October 14, 2011.

[629] M. Merkle, Lj. Petrovic. On Schur-convexity of some distribution functions. Publ. Inst.Math. 56(70):111–118, 1994.

[630] Milan Merkle. Logarithmic convexity and inequalities for the Gamma function. J. Math.Anal. Appl 203:369–380, 1996.

[631] Milan J. Merkle. Mills ratio for the Gamma distribution. Univ. Beograd. Publ. Elektrotehn.

Fak. Ser. Mat. Fiz. [Univerzitet u Beogradu. Publikacije Elektrotehnickog Fakulteta. SerijaMatematika i Fizika] 68–70(678–715), 1980.

[632] Milan J. Merkle. Some inequalities for the χ2 distribution function. Univ. Beograd. Publ.

Elektrotehn. Fak. Ser. Mat. [Univerzitet u Beogradu. Publikacije Elektrotehnickog Fakulteta.Serija Matematika] 2, 1991.

[633] Milan J. Merkle. Some inequalities for the χ2 distribution function and the exponential

function. Arch. Math. (Basel) [Archiv der Mathematik. Archives of Mathematics. ArchivesMathematiques] 60(5):451–458, 1993.

[634] M. Meyer, A. Pajor. Sections of the unit ball of lnp . J. of Funct. Anal. 80(1):109–123, 1988.

[635] Ion Mihoc. On the measures of information of order α associated with some probabilitydistributions. Studia Univ. Babes Bolyai Math. [Universitatis Babes Bolyai. Studia. Math-

ematica] 35(3):65–76, 1990.

[636] Zarko Mijajlovic. On some formulas involving !n and the verification of the !n-hypothesis

by use of computers. Publ. Inst. Math 47(61):24–32, 1990.

[637] M. Mikolas. Uber die Beziehung zwischen der Gammafunktion un den Trigonometrischen

Funktionen. Acta Math. Acad. Sci. Hung. 4:143–157, 1953. Fasc.1–2.

[638] M. Mikolas. Zur Theorie der Gammafunction, der Riemannschen Zetafunktion und ver-wandter Funktionen, I. Acta Math. Acad. Sci. Hung. 6:381–438, 1955. Fasc.3–4.

[639] C. Miller. An extension of Holder’s theorem on the Gamma function. Israel J. Math. 97:183–

187, 1997.[640] L.M. Milne-Thomson. The calculus of finite differences. Macmillan, London, 1960.

[641] John Milnor. On polylogarithms, Hurwitz Zeta functions, and the Kubert identities. En-

seign. Math. (2) [L’Enseignement Mathematique. Revue Internationale. IIe Serie] 29(3–4):281–322, 1983.

[642] G. V. Milovanovic. Expansions of the Kurepa function. Publ. Inst. Math 57(71):81–90, 1995.

[643] G.V. Milovanovic. Recent progress in inequalities. Kluwer Acad. Publ., 1998.

[644] D.S. Mitrinovic, J.E. Pecaric. Note on the Gauss-Winkler inequality. Anzeiger Oster. Akad.

Wiss. Math.-Nat. Kl.(6):89–92, 1986.[645] Daniel S. Moak. The q-Gamma function for q > 1. Aequationes Math. [Aequationes Mathe-

maticae] 20(2–3):278–285, 1980.

[646] Daniel S. Moak. The q-Gamma function for x < 0. Aequationes Math. [Aequationes Math-ematicae] 21(2–3):179–191, 1980.

[647] Daniel S. Moak. The q-analogue of the Laguerre polynomials. J. Math. Anal. Appl. [Journal

of Mathematical Analysis and Applications] 81(1):20–47, 1981.[648] Daniel S. Moak. The q-analogue of Stirling’s formula. Rocky Mountain J. Math. [The Rocky

Mountain Journal of Mathematics] 14(2):403–413, 1984.

[649] D.S. Moak. The q-Gamma function for x < 0. Aeq. Math. 21(2–3):179–191, 1980.[650] E. I. Moiseev. An inequality connected with the solution of a differential equation. Differ-

entsialnye Uravneniya 22(1):82–94, 182, 1986. Russian.[651] R. E. M. Moore, I. O. Angell. Voronoi polygons and polyhedra. J. Comput. Phys. [Journal

of Computational Physics] 105(2):301–305, 1993.

[652] Carlos Julio Moreno. The Chowla-Selberg formula. J. Number Theory [Journal of NumberTheory] 17(2):226–245, 1983.

[653] Y. Morita. A p-adic integral representation of the p-adic l-function. J. Reine Angew. Math.

302:71–95, 1978.[654] Yasuo Morita. A p-adic analogue of the Gamma function. J. Fac. Sci. Univ. Tokyo 22:255–

266, 1975.


[655] Cristinel Mortici. Estimating Gamma function by Digamma function. Mathematical andComputer Modelling 52(5–6):942–946, 2010. Submitted by. Gergo Nemes, July 20, 2010.

[656] Cristinel Mortici. New approximation formulas for evaluating the ratio of Gamma functions.

Mathematical and Computer Modelling 52(1-2):425–433, 2010. Submitted by. Gergo Nemes,July 20, 2010.

[657] Cristinel Mortici. New approximations of the Gamma function in terms of the Digamma

function. Applied Mathematics Letters 23(1):97–100, 2010. Submitted by. Gergo Nemes,July 20, 2010.

[658] Martin E. Muldoon. Some Characterizations of the Gamma Function involving the Notionof Complete Monotonicity. Aequationes-Math 8:212–215, 1978.

[659] Martin-E. Muldoon. Some Monotonicity Properties and Characterization of the Gamma

Function. Aequationes-Math. 18:54–63, 1978.[660] M. Muller, D. Schleicher. Fractional sums and Euler-like identities. Preprint, 2005.

[661] M. Muller, D. Schleicher. How to add a non-integer number of terms, and how to produce

unusual infinite summations. J. Comp. Appl. Math. 178:347–360, 2005.[662] Hajime Nakazato. The q-analogue of the p-adic Gamma function. Kodai Math. J. [Kodai

Mathematical Journal] 11(1):141–153, 1988.

[663] V. Namias. A simple derivation of Stirling’s asymptotic series. Amer. Math. Monthly 93:25–29, 1986.

[664] S. B. Nandi. On the exact distribution of a normalized ratio of the weighted geometric mean

to the unweighted arithmetic mean in samples from Gamma distributions. J. Amer. Statist.Assoc. [Journal of the American Statistical Association] 75(369):217–220, 1980.

[665] C. Nasim. An integral equation involving Fox’s h-function. Indian J. Pure Appl. Math.

[Indian Journal of Pure and Applied Mathematics] 13(10):1149–1162, 1982.[666] T. Negoi. A quicker convergence to Euler’s constant (Romanian). GMA (Bucuresti)

15(2):111–113, 1997.[667] J. E. Nelson, T. Regge. Quantization of 2 + 1 gravity for Genus 2. Phys.Rev.D (3) [Physical

Review. D. Third Series] 50(8):5125–5129, 1994.

[668] Peter R. Nelson, Edward J. Dudewicz, Aydin Ozturk, Edward C. van der Meulen, editors.The frontiers of statistical computation, simulation and modeling., volume 25 of American

Sciences Press Series in Mathematical and Management Sciences. American Sciences Press,

vi+338 pp, Columbus, OH, 1991. Proceedings of the First International Conference onStatistical Computing (ICOSCO-I) held in Cesme, March 30–April 2, 1987. Vol. I.

[669] Gergo Nemes. On the Coefficients of the Asymptotic Expansion of n!. Journal of Integer

Sequences 13(6):1–5, 2010. Submitted by. Gergo Nemes, July 20, 2010.[670] Gergo Nemes. Asymptotic expansion for logn! in terms of the reciprocal of a triangular

number. Acta Mathematica Hungarica 129(3):254–262, 2010. Submitted by. Milan Merkle,

October 14, 2011.[671] Gergo Nemes. New asymptotic expansion for the Gamma function. Archiv der Mathematik

95(2):161–169, 2010. Submitted by. Milan Merkle, October 14, 2011.[672] Gergo Nemes. More accurate approximations for the gamma function. Thai Journal of

Mathematics 9(1):21–28, 2011. Submitted by. Milan Merkle, October 14, 2011.

[673] Yu.V. Nesterenko. Modular functions and transcendence questions. Mat. Sb. 187:65–96,1996.

[674] E. Neuman. Ineqalities involving a logarithmically convex function and their applications to

special functions. J. Ineq. Pure Appl. Math. 7, 2006.[675] T.A. Newton. Derivation of a factorial function by method of analogy. Amer. Math. Monthly

68:917–920, 1961.[676] S. Niizeki, M. Sasato, H. M. Srivastava. Further remarks on the integral of exp(−x2) over R.

Mem. Fac. Sci. Kochi Univ. Ser. A Math. 19:97–102, 1998. Submitted by. H. M. Srivastava,

2. May 2010.

[677] A. Nikiforov, V. Ouvarov. Elements de la theorie des fonctions speciales. Ed. Mir, Moscou,1976.

[678] A. I. Nikishov, V. I. Ritus. The width of Stokes lines. Original: Teoret. Mat. Fiz. [Rossi-iskaya Akademiya Nauk. Teoreticheskaya i Matematicheskaya Fizika] 92(1):24–40, 1992.

[679] A. I. Nikishov, V. I. Ritus. The width of Stokes lines. Translation: Theoret. and Math. Phys.

[Theoretical and Mathematical Physics] 92(1):711–721, 1993.


[680] Katsuyuki Nishimoto, Ram Kishore Saxena. An application of Riemann-Liouville operatorin the unification of certain functional relations. J. College Engrg. Nihon Univ. Ser. B

[Nihon University. College of Engineering. Journal. Series-B] 32:133–139, 1991.

[681] M. Nishizawa. On a q-analogue of the multiple Gamma functions. Lett. Math. Phys.37(2):201–209, 1996.

[682] R. H. Norden. On the distribution of the time to extinction in the stochastic logistic popu-

lation model. Adv. in Appl. Probab. [Advances in Applied Probability] 14(4):687–708, 1982.[683] J. Nunemacher. On computing Euler’s constant. Math. Mag. 65:313–322, 1992.

[684] Arthur Ogus. A p-adic analogue of the Chowla-Selberg formula. In Collection: p-adic anal-ysis (Trento, 1989), volume 1454 of Lecture Notes in Math, 319–341, Berlin, 1990. Springer.

[685] M. O. Ojo. A note on the convolution of logistic random variables. Sankhya Ser. A [Sankhya

(Statistics). The Indian Journal of Statistics. Series A] 48(3):409–410, 1986.[686] G. O. Okikiolu. Weighted inequalities for convolutions and homogeneous convolutions. Bull.

Math. [George O. Okikiolu. Bulletin of Mathematics](1):48–84, 1981–1982.

[687] A.B. Olde Daalhuis. Asymptotic expansions for q-Gamma, q-exponential, and q-Bessel func-tions. J. Math. Anal. Appl. 186(3):896–913, 1994.

[688] F.W.G. Oler. Asymptotics and special functions. Academic Press, New York–London, 1974.

[689] I. Olkin. An inequality satisfied by the Gamma function. Skand. Aktuarietidiskr 61:37–39,1959.

[690] F. W. J. Olver. Uniform, exponentially improved, asymptotic expansions for the general-

ized exponential integral. SIAM J. Math. Anal. [SIAM Journal on Mathematical Analysis]22(5):1460–1474, 1991.

[691] F. W. J. Olver. Asymptotic expansions of the coefficients in asymptotic series solutions of

linear differential equations. Methods Appl. Anal. [Methods and Applications of Analysis]1(1):1–13, 1994.

[692] B. Osgood, R. Phillips, P. Sarnak. Extremals of determinants of laplacians. J. Funct.Anal.80:148–211, 1988.

[693] A. Ostrowski. On the remainder term of the Euler-Maclaurin formula. J. Reine Angew.

Math. 239/240:268–286, 1970.[694] A. Palanques Mestre, P. J. Pascual Broncano. The 1/nf expansion of the γ and β functions

in Q.E.D. Comm. Math. Phys. [Communications in Mathematical Physics] 95(3):277–287,

1984.[695] L. Panaitopol. Refinement of Stirling’s formula (Romanian). Gaz. Mat. (Bucuresti) 9:329–

332, 1985.

[696] John Panaretos. Unique properties of some distributions and their applications. Statistica(Bologna) 41(4):567–574, 1981.

[697] R. B. Paris. Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput.

Appl. Math. [Journal of Computational and Applied Mathematics] 41(1–2):117–133, 1992.Asymptotic methods in analysis and combinatorics.

[698] R. B. Paris, Alastair D. Wood. Exponentially-improved asymptotics for the Gamma func-tion. J. Comput. Appl. Math. [Journal of Computational and Applied Mathematics] 41(1–

2):135–143, 1992. Asymptotic methods in analysis and combinatorics.

[699] V. Pasquier, M. Gaudin. L’Ansatz de Bethe pour les representations cycliques des groupesquantiques. [the Bethe Ansatz for cyclic representations of quantum groups]. In Collection:

R.C.P. 25, (Strasbourg, 1992), volume 44 of Prepubl. Inst. Rech. Math. Av, 1993/41, 163–

174. Univ. Louis Pasteur, Strasbourg, 1993. French.[700] P. I. Pastro. Analytic characterization of the q-Gamma function. Riv. Mat. Univ. Parma

(4) [Rivista di Matematica della Universita di Parma. Serie IV] 9:387–390, 1983. Italian.[701] P. I. Pastro. The q-analogue of Holder’s theorem for the Gamma function. Rend. Sem. Mat.

Univ. Padova [Rendiconti del Seminario Matematico dell’Universita di Padova] 70:47–53,

1983.

[702] G. P. Patil, M. V. Ratnaparkhi. Characterizations of certain statistical distributions basedon additive damage models involving Rao-Rubin condition and some of its variants. Sankhya

(Statistics). The Indian Journal of Statistics. Series B 39(1):65–75, 1977.[703] J Pecaric, J. Sandor. On certain inequalities involving euler gamma and digamma functions.

Octogon Math. Mag. 16(2):946–948, 2008.


[704] A. Peretti. A transcendent exact differential equation for the logarithm of the Gammafunction. Bull. Number Theory Related Topics [Bulletin of Number Theory and Related

Topics] 10(2):22–25, 1986.

[705] A. Peretti. The irrationality of Euler’s constant, III. Notes Numb. Th. Discr. Math.3(4):183–193, 1997.

[706] O. Perron. Die Lehre von den Kettenbruche, volume 2. Teubner, Stuttgart, 3rd edition,

1957.[707] J. Pecaric, G. Allasia, C. Giordano. Convexity and the Gamma function. Indian J. Math.

41:79–93, 1999.[708] S. Pincherle. Sur une generalisation des fonctions Euleriennes. Comptes Rendues 106:265–

268, 1888.

[709] M. Piros. A generalization of the Wallis formula. Miskolc Math. Notes 4(2):151–155, 2003.[710] T. K. Pogany, H. M. srivastava. Some improvements over Love’s inequality for the Laguerre

function. Integral Transforms Spec. Funct. 18:351–358, 2007. Submitted by. H. M. Srivas-

tava, 2. May 2010.[711] T. K. Pogany, H. M. Srivastava, Z. Tomovski. Some families of Mathieu a-series and al-

ternating Mathieu a-series. Appl. Math. Comput. 173:69–108, 2006. Submitted by. H. M.

Srivastava, 2. May 2010.[712] B.-B. Pokhodzei. Beta and Gamma methods of simulating binomial and Poisson distri-

butions. Zh. Vychisl. Mat. i Mat. Fiz. [Akademiya Nauk SSSR. Zhurnal Vychislitelnoi

Matematiki i Matematicheskoi Fiziki] 24(2):187–193, 1984. Russian.[713] G. Polya, G. Szego. Problems and theorems in analysis, volume I. Springer Verlag, 1972/78.

Reprint 1998.

[714] S. Ponnusamy, M. Vuorinen. Asymptotic expansions and inequalities for hypergeometricfunctions. Mathematika 44:278–301, 1997.

[715] E.L. Post. The generalized Gamma functions. Annals of Math. 20:202–217, 1918/19.[716] Vasile Preda. A characterization by factorization of the Gamma distribution. An. Univ.

Bucuresti Mat. [Analele Universitatii Bucuresti. Matematica] 36:68–69, 1987.

[717] Vasile C. Preda. Sur la distribution Gamma et le principe du maximum de l’information.[On the Gamma distribution and the maximum principle of information]. Bull. Math. Soc.

Sci. Math. R. S. Roumanie (N.S.) [Bulletin Mathematique de la Societe des Sciences Math-

ematiques de la Republique Socialiste de Roumanie. Nouvelle Serie] 31(4):355–358, 1987.French.

[718] A. Prekopa, R. J. B. Wets, editors. Stochastic programming 84. II. North Holland Publishing

Co. pp. i–vii and 1–181., Amsterdam–New York, 1986. Math. Programming Stud. No. 28(1986).

[719] A. I. Prieto, J. Matera, H. M. Srivastava. Use of the generalized Lommel-Wright function

in a unified presentation of the Gamma-type functions occurring in diffraction theory andassociated probability distributions. Integral Transforms Spec. Funct. 17:365–378, 2006.Submitted by. H. M. Srivastava, 2. May 2010.

[720] A. Pringsheim. Zur Theorie der Gammafunktion. Math. Ann. (Journal fur Math.) 31:455–

481, 1888.

[721] Helmut Prodinger. How to select a loser. Discrete Math. [Discrete Mathematics] 120(1-3):149–159, 1993.

[722] A.P. Prudnikov et al. Integrals and Series (Supplementary Chapters). Nauka, Moscow, 1986.

See also: Integrals and Series, vol.22, Special functions (trans. from the Russian by N.M.Queen) end ed., Gordon and Breach, New York, 1988., Russian.

[723] F.E. Prym. Zur Theorie der Gammafunktion. J. Reine Angew. Math. 82:165–172, 1877.[724] F. Qi. The best bound in Kershaw’s inequality and two completely monotonic functions.

RGMIA Research Report Collection 9(4), 2006.

[725] F. Qi. A class of logaritmically completely monotonic functions and application to the best

bounds in the second Gautschi-Kershaw’s inequality. RGMIA Research Report Collection9(4), 2006.

[726] F. Qi, C.P. Chen. A new proof of the best bounds in Wallis’ inequality. RGMIA ResearchReport Collection 6(2), 2003.

[727] F. Qi, C.P. Chen. A complete monotonicity of the Gamma function. RGMIA Research

Collection 7(1), 2004.


[728] F. Qi, Wei Li. Two logarithmically completely monotonic functions connected with theGamma function. RGMIA Research Report Collection 8(3), 2005.

[729] F. Qi, J.Q. Mei. Some inequalities of the incomplete Gamma and related functions. Z. Anal.

Anwendungen 18(3):793–799, 1999.[730] Feng Qi. Bounds for the Ratio of two Gamma Functions. J. Ineq. Appl. 2010, 2010. Sub-

mitted by. Feng Qi, 27. April 2010, Article ID 493058, 84 pages, doi:10.1155/2010/493058.

[731] S.L. Qiu, M. Vuorinen. Some properties of the Gamma and Psi functions, with applications.Math. Comp. 74:723–742, 2005.

[732] C. Radhakrishna, A. V. Dattatreya Rao, G. V. S. R. Anjaneyulu. Estimation of parame-ters in a two-component mixture generalized Gamma distribution. Comm. Statist. Theory

Methods [Communications in Statistics. Theory and Methods] 21(6):1799–1805, 1992.

[733] Mizan Rahman. Some infinite integrals of q-Bessel functions. In Collection: RamanujanInternational Symposium on Analysis (Pune, 1987), 117–137, New Delhi, 1989. Macmillan

of India.

[734] E.D. Rainville. Special functions. Chelsea Publ. Company, New York, 1960.[735] B Raja Rao. Gautschi-type inequalities for incomplete absolute moments of a multivariate

distribution with applications. Metron 39(3–4):43–51, 1981.

[736] B. Raja Rao, K. G. Janardan, D. J. Schaeffer. A Lagrangian Gamma distribution of thesecond kind. Biometrical J. [Biometrical Journal] 26(8):941–946, 1984.

[737] S. Ramanujan. Some definite integrals. Messenger of Math. 44:10–18, 1915. See also Col-

lected papers of Srinivasa Ramanujan, ed. by G.H. Hardy, P.V. Seshu Aiyar and B.M.Wilson, Cambridge Univ. Press, 1927; reprinted by Chelsea, New York, 1962.

[738] S. Ramanujan. The lost notebook and other unpublished manuscripts. Springer-Verlag,

Berlin, 1988.[739] B. R. Rao. An improved inequality satisfied by the Gamma function. Skand.

Aktuarietidiskr:78–83, 1969.[740] B. R Rao. On an inequality satisfied by the Gamma function and its logarithmic derivatives.

Metron 39(3–4):125–131, 1981.

[741] B. R. Rao. On some analogs of Rao-Cramer inequality and a series of inequalities satisfiedby the Gamma function. Metron 42(1–2):89–108, 1984.

[742] B. R. Rao, M. L. Garg. An inequality for the Gamma function using the analogue of Bhat-

acharyya’s inequality. Skand. Aktuarietidiskr:71–77, 1969.[743] G. Rash. Notes on the Gamma-funktion. Ann. of Math. (2) 32:591–599, 1931.

[744] Th. M. Rassias, H. M. Srivastava. Some classes of infinite series associated with the Riemann

Zeta and Polygamma functions and generalized harmonic numbers. Appl. Math. Comput.131:593–605, 2002. Submitted by. H. M. Srivastava, 2. May 2010.

[745] M. V. Ratnaparkhi. Some bivariate distributions of (x, y) where the conditional distribution

of y, given x, is either Beta or unit-Gamma. In Collection: Statistical distributions inscientific work, Vol. 4 (Trieste, 1980), volume 79 of NATO Adv. Study Inst. Ser. C: Math.Phys. Sci, 389–400, Dordrecht–Boston, Mass, 1981. Reidel.

[746] Robert R. Read. Representation of certain covariance matrices with application to asymp-

totic efficiency. J. Amer. Statist. Assoc. [Journal of the American Statistical Association]

76(373):148–154, 1981.[747] Axel Reich. On Hypertranscendental functions. In Collection: Topics in classical number

theory, Vol. I, II (Budapest, 1981), Colloq. Math. Soc. Janos Bolyai, 34, 1349–1369. North-

Holland, Amsterdam-New York, 1984.[748] Iris Reinhard. The quantitative behaviour of some slowly growing population-dependent

Markov branching processes. In Collection: Stochastic modelling in biology (Heidelberg,1988), 267–277, Teaneck, NJ, 1990. World Sci. Publishing.

[749] R. Remmert. Classical topics in complex function theory. Springer-Verlag, 1998.

[750] Reinhold Remmert. Wielandt’s Theorem about the γ-function. Amer. Math. Monthly [The

American Mathematical Monthly] 103(3):214–220, 1996.[751] K. J. A. Revfeim. Approximation for the cumulative and inverse Gamma distribution.

Statist. Neerlandica [Statistica Neerlandica. Quarterly Journal of the Netherlands Societyfor Statistics and Operations Research] 45(3):327–331, 1991.

[752] E. Ya. Riekstynsh. Asimptotika i otsenki nulei nekotorykh spetsialnykh funktsii. [Asymptotic

behavior of and estimates for the zeros of some special functions], volume 122 of LAFI. Akad.Nauk Latv. SSR, Inst. Fiz., 40 pp, Salaspils, 1987. With an English summary., Russian.


[753] H. Robbins. A remark on Stirling’s formula. Amer. Math. Monthly 62:26–29, 1955.[754] Bertram Ross. The early development of the Gamma function. Ganita Bharati [Ganita

Bharati. Indian Society for History of Mathematics. Bulletin] 2(1–2):1–15, 1980.

[755] Bertram Ross. Methods of Summation. Descartes Press Co, x+127 pp., Koriyama, Japan,1987.

[756] E. Rouche. Sur la formule de Stirling. Comptes Rendus 110:513–515, 1890.

[757] M. Z. Rovinskii. Meromorphic functions connected with polylogarithms. Funktsional. Anal. iPrilozhen. [Akademiya Nauk SSSR. Funktsionalnyi Analiz i ego Prilozheniya] 25(1):88–91,

1991.[758] M. Z. Rovinskii. Meromorphic functions connected with polylogarithms. Functional Anal.

Appl. [Functional Analysis and its Applications] 25(1):74–77, 1991.

[759] Ch.H. Rowe. A proof of the asymptotic series for log γ(z) and log γ(z + 1). Ann. of Math.(2) 32:10–16, 1931.

[760] Dilip Roy. A characterization of the generalized Gamma distribution. Calcutta Statist. As-

soc. Bull. [Calcutta Statistical Association Bulletin] 33(131–132):137–141, 1984.[761] Dilip Roy. Addendum to: “A characterization of the generalised Gamma distribution”.

Calcutta Statist. Assoc. Bull. [Calcutta Statistical Association Bulletin] 34(133–134):125,

1985.[762] Dilip Roy, S. P. Mukherjee. Generalised mixtures of exponential distributions. J. Appl.

Probab. [Journal of Applied Probability] 25(3):510–518, 1988.

[763] T. Royen. On some multivariate Gamma-distributions connected with spanning trees. Ann.Inst. Statist. Math. [Annals of the Institute of Statistical Mathematics] 46(2):361–371, 1994.

[764] Thomas Royen. Multivariate Gamma distributions with one-factorial accompanying cor-

relation matrices and applications to the distribution of the multivariate range. Metrika[Metrika. International Journal for Theoretical and Applied Statistics] 38(5):299–315, 1991.

[765] Thomas Royen. On representation and computation of multivariate Gamma distributions.In Collection: Data analysis and statistical inference, 201–216. Eul, Bergisch Gladbach,

1992.

[766] Lee A. Rubel. A survey of transcendentally transcendental functions. Amer. Math. Monthly[The American Mathematical Monthly] 96(9):777–788, 1989.

[767] H. Ruben. Variance bounds and orthogonal expansions in Hilbert space with an application

to inequalities for Gamma function and π. J.Reine und agnew Math 225:147–153, 1967.[768] H. Ruben. Variance bounds and orthogonal expansions in Hilbert space with an application

to inequalities for Gamma functions and π. Journal fur Mathematik Band 225:19–25, 1967.

[769] W. Rudin. Principles of mathematical analysis. McGraw-Hill, New York, 1953.[770] A.A. Samoletov. A sum containing factorials. J. Comput. Appl. Math. 131:503–504, 2001.

[771] J. Sandor. On a property of the harmonic series (Romanian). Gaz. Mat. (Bucuresti)

93(8):311–312, 1988.[772] J. Sandor. Remark on a function which generalizes the harmonic series. C.R. Acad. Bulg.

Sci. 41(5):19–21, 1988.[773] J. Sandor. Sur la fonction Gamma. Publ. C.R.M.P. Neuchatel, Ser. I 21:4–7, 1989. Sub-

mitted by. Jozsef Sandor, February 3, 2010.

[774] J. Sandor. On the Gamma function II. Publ. Centre Rech. Math. Pures Chambery:10–12,1997. Serie 1, Fasc. 28.

[775] J. Sandor. On Stirling’s formula (Romanian). Lucr. Semin. Did. Mat. (Cluj) 14:235–240,

1998.[776] J. Sandor. On convex functions involving Euler’s Gamma function. Octogon Math. Mag.

8:514–516, 2000.[777] J. Sandor. Two theorems on convexity or concavity of functions. Octogon Math. Mag.

8(2):337–339, 2000.

[778] J. Sandor. On the Gamma function III. Centre Rech. Math. Pures Neuchatel 19:33–40,

2001. Serie 2.[779] J. Sandor. On a limit involving the Euler Beta-function. Octogon Math. Mag. 11(1):251–252,

2003.[780] J. Sandor. On a limit involving the Euler Gamma function. Octogon Math. Mag. 11(1):262–

264, 2003.

[781] J. Sandor. On a property of the Gamma function. Octogon Math. Mag. 11(1):198–109, 2003.


[782] J. Sandor. On certain inequalities for the ratios of Gamma functions. Octogon Math. Mag.12(2B):1052–1059, 2004.

[783] J. Sandor. A note on certain inequalities for the Gamma function. J. Ineq. Pure Appl. Math.

6(3), 2005.[784] J. Sandor. On certain limits involving the Euler constants e and γ. Octogon Math. Mag.

13(1A):358–361, 2005.

[785] J. Sandor. On certain inequalities for the Gamma function. RGMIA Research Report Col-lection 9(1), 2006.

[786] J. Sandor. Generalized Euler constants and Schlomilch-Lemonnier type inequalities. J. Math.Anal. Appl. 328(2):1336–1342, 2007.

[787] J. Sandor. A note on certain euler-mascheroni type sequences. Sci. Magna 4(1):60–62, 2008.

[788] J. Sandor. Selected chapters of geometry, analysis and number theory: classical topics innew perspectives. LAP Lambert Academic Publishers, Koln, Germany, 2009.

[789] J. Sandor, B. Crstici. Handbook of number theory, II. Springer-Verlag, 2005.

[790] J. Sandor, L. Debnath. On certain inequalities involving the constant e and their applica-tions. J. Math. Anal. Appl. 249:569–582, 2000.

[791] J. Sandor, D.S. Mitrinovic, B. Crstici. Handbook of number theory I. Springer Verlag, 2006.

[792] J. Sandor, L. Toth. A remark on the Gamma function. Elem. Math. 44(3):73–76, 1989.[793] J. Sandor, L. Toth. On some arithmetical products. Publ. Centre Rech. Mat. Pures,

Neuchatel, Serie I 20:5–8, 1990.

[794] Jozsef Sandor. Remark on a function which generalizes the harmonic series. C. R. Acad.Bulgare Sci. [Doklady Bolgarskoi Akademii Nauk. Comptes Rendus de l’Academie Bulgare

des Sciences] 41(5):19–21, 1988.

[795] Jozsef Sandor. On some Diophantine Equations involving the Factorial of a Number., vol-ume 21 of Seminar Arghiriade. Universitatea din Timisoara, Facultatea de Stiinte ale Na-

turii, Sectia Matematica, 4 pp., Timisoara, 1989.[796] G. Sansone, J. Gerretsen. Lectures on the theory of functions of a complex variable. Noord-

hoff, Groningen, 1960.

[797] Peter Sarnak. Determinants of Laplacians. Comm. Math. Phys. [Communications in Math-ematical Physics] 110(1):113–120, 1987.

[798] Z. Sasvari. An elementary proof of Binet’s formula for the Gamma function. Amer. Math.

Monthly 106(2):156–158, 1999.[799] Y. S. Sathe, U. J. Dixit. A comment on: “A characterization of the generalised Gamma

distribution” [Calcutta Statist. Assoc. Bull. 33 (1984), no. 131, 137–141] by D. Roy. Calcutta

Statist. Assoc. Bull. [Calcutta Statistical Association Bulletin] 34(133–134):123–124, 1985.[800] Dominic Y. Savio, Edmund A. Lamagna, Shing Min Liu. Summation of harmonic numbers.

In Collection: Computers and mathematics (Cambridge, MA, 1989), 12–20. Springer, New

York–Berlin, 1989.[801] Ram Kishore Saxena, Jeta Ram. Some results of generalized hypergeometric functions with

applications. J. Indian Acad. Math. [Indian Academy of Mathematics. Journal] 14(1):53–59, 1992.

[802] Michele Sce. Sugli sviluppi di Charlier. [On the Charlier expansions], volume 200 of Pub-

blicazioni Serie III [Publication Series III]. Istituto per le Applicazioni del Calcolo “MauroPicone” (IAC), 54 pp, Rome, 1979. Italian.

[803] R. Schark. Konstanten in der Mathematik—Variabel Betrachtet. [Constants in

Mathematics—Variously Considered], volume 76 of Deutsch-Taschenbucher [Deutsch Pa-perbacks]. Verlag Harri Deutsch, vi+188 pp, Thun, 1992. In German.

[804] H. J. Schell. Gleichmassige Asymptotische Entwicklungen fur Parameterintegrale mit Zweireellen Parametern. [Uniform asymptotic expansions of parameter i]. Z. Anal. Anwendungen[Zeitschrift fur Analysis und Ihre Anwendungen] 2(2):159–173, 1983. German.

[805] Walter Schempp. Approximation und Transformationsmethoden. II. In Collection: Numer-

ical methods of approximation theory, Vol. 5 (Conf., Math. Res. Inst., Oberwolfach, 1979),volume 52 of Internat. Ser. Numer. Math., 277–281. Birkhauser, Basel-Boston, Mass., 1980.

German.[806] Hans Schenkel. Kritisch-Historische Untersuchung uber die Theorie der Gammafunction

und Euler’schen Integrale. Uster-Zurich, 1894. Inaugural Dissertation, German.


[807] W. H. Schikhof. Ultrametric Calculus. Cambridge Studies in Advanced Mathematics, 4.Cambridge University Press viii+306 pp, $39.50., Cambridge-New York, 1984. An introduc-

tion to p-adic analysis.

[808] O. Schlomilch. Uber den Quotienten zweier Gammafunktionen. Zeitschrift fur Math. u.Physik 25:351–352, 1880.

[809] Bruce W. Schmeiser, Ram Lal. Bivariate Gamma random vectors. Oper. Res. [Operations

Research. The Journal of the Operations Research Society of America] 30(2):355–374, 1982.[810] C. G. Schmidt. Gauss sums and the classical γ-function. Bull. London Math. Soc. [The

Bulletin of the London Mathematical Society] 12(5):344–346, 1980.

[811] Eric Schmutz. Asymptotic expansions for the coefficients of eP (z). Bull. London Math. Soc.[The Bulletin of the London Mathematical Society] 21(5):482–486, 1989.

[812] J.A. Schobloch. Uber Beta und Gammafunktionen. Halle, 1884.[813] I. J. Schoenberg. On Polya frequency functions, I. The totally positive functions and their

Laplace Transforms. J. Analyse Math 1:331–374, 1951.

[814] Reinhard Schuster. A generalization of the Barnes g-function. Z. Anal. Anwendungen[Zeitschrift fur Analysis und ihre Anwendungen] 11(2):229–236, 1992.

[815] Christoph J. Scriba. Von Pascals Dreieck zu Eulers Gamma-Funktion. Zur Entwicklung

der Methodik der Interpolation. [From Pascal’s triangle to Euler’s Gamma function]. InCollection: Mathematical perspectives, 221–235. Academic Press, New York-London, 1981.

German.

[816] James B. Seaborn. Hypergeometric Functions and their Applications., volume 8 of Texts inApplied Mathematics. Springer Verlag, xiv+250 pp, $39.00, New York, 1991.

[817] A. Selberg. Bemerkinger on et multipelt integral. Norsk. Mat. Tidsskr. 26:71–78, 1944.

[818] R.E. Shafer. Inequality on ψ(x), solution to problem 6456. Amer. Math. Monthly 92:598,1985.

[819] D. N. Shanbhag, M. Sreehari. An extension of Goldie’s result and further results in infi-nite divisibility. Z. Wahrsch. Verw. Gebiete [Zeitschrift fur Wahrscheinlichkeitstheorie und

Verwandte Gebiete] 47(1):19–25, 1979.

[820] R. Shantaram. Further stronger Gamma function inequalities. Skand. Aktuarietidiskr.:204–206, 1968.

[821] Kamran Sharifi, Alberto Leon-Garcia. Estimation of Shape Parameter for generalized Gauss-

ian Distributions in subband distribution of video. IEEE Trans. Circ. Syst. Video. Techn5(1):52–55, 1995.

[822] L.R. Shenton, K.O. Bonne. Continued fractions for the Psi function and its derivatives.

SIAM J. Appl. Math. 20(4):547–554, 1971.[823] Wei Kei Shiue, Lee J. Bain. A two-sample test of equal Gamma distribution scale parameters

with unknown common shape parameter. Technometrics. A Journal of Statistics for the

Physical, Chemical and Engineering Sciences 25(4):377–381, 1983.[824] Wei Kei Shiue, Lee J. Bain, Max Engelhardt. Test of equal Gamma-distribution means with

unknown and unequal shape parameters. Technometrics. A Journal of Statistics for thePhysical, Chemical and Engineering Sciences 30(2):169–174, 1988.

[825] Masaaki Sibuya, Ryoichi Shimizu. The generalized hypergeometric family of distributions.

Ann. Inst. Statist. Math. [Annals of the Institute of Statistical Mathematics] 33(2):177–190,1981.

[826] C. L. Siegel. uber Die Analytische Theorie Der Quadratischen Formen. Ann. of Math. (2)

36(2):527–606, 1935. In German.[827] Chiaw Hock Sim. Characterization of the Gamma distribution. Bull. Malaysian Math. Soc.

(2) [Malaysian Mathematical Society. Bulletin. Second Series] 9(2):81–84, 1986.

[828] A. K. Singh, Y. N. Prasad. Basic properties of the transform involving an h-function ofr variables as kernel. J. Indian Acad. Math. [Indian Academy of Mathematics. Journal]

4(2):94–100, 1982.[829] Housila Prasad Singh. Estimation of common parameter in k Gamma distribution. Gujarat

Statist. Rev. [Gujarat Statistical Review] 15(1):45–48, 1988.

[830] Prahlad Singh Kaurav. Incomplete Gamma-function as a generating function of an,v(x). Vi-jnana Parishad Anusandhan Patrika (The Research Journal of the Hindi Science Academy)

36(3):197–200, 1993. Hindi.

[831] Dusan Slavic. Tables for functions γ(x) and 1/γ(x). Univ. Beograd Publ. Elektrotehn. Fak.Ser. Mat. Fiz 357–380:69–74, 1971.


[832] S. P. Smith, K. Hammond. Rank regression with log Gamma residuals. Biometrika75(4):741–751, 1988.

[833] Milton Sobel, Martin Wells. Dirichlet integrals and moments of Gamma distribution order

statistics. Statist. Probab. Lett. [Statistics and Probability Letters] 9(5):431–437, 1990.[834] A. Sofo, H. M. Srivastava. A family of sums containing factorials. Integral Transforms Spec.

Funct. 20:393–399, 2009. Submitted by. H. M. Srivastava, 2. May 2010.

[835] D. Somasundaram. Srinivasa Ramanujan and Gamma function. Math. Ed. [MathematicalEducation] 4(1):17–22, 1987.

[836] J.W. Son, D.S. Jang. On q-analogues of Stirling series. Commun. Korean Math. Soc.14(1):57–68, 1999.

[837] J. Sondow. Criteria for irrationality of Euler’s constant. Proc. Amer. M.S. 131(11):3335–

3344, 2003. (Electronic).[838] G.W. Soules. New permanental upper bounds for nonnegative matrices. Linear and Multi-

linear Algebra 51(4):319–337, 2003.

[839] G.W. Soules. Permanental bounds for nonnegative matrices via decomposition. Linear Al-gebra Appl. 394:73–89, 2005.

[840] J. Spanier, K.B. Oldham. An atlas of functions. Springer Verlag, Berlin, 1987.

[841] R. Spira. Calculation of the Gamma function by Stirling’s formula. Math. Comp.25(114):317–322, 1971.

[842] J.L. Spouge. Computation of the Gamma, Digamma and Trigamma functions. SIAM J.

Numer. Anal. 31:931–944, 1994.[843] G. K. Srinivasan. The Gamma function: An Eclectic Tour. Amer. Math. Monthly 114:297–

315, 2007. Submitted by. Christian Berg, 4. May 2010.

[844] H. M. Srivastava. Some elliptic integrals of Barton and Bushell. J. Phys. A: Math. Gen.28:2305–2312, 1995. Submitted by. H. M. Srivastava, 2. May 2010.

[845] H. M. Srivastava. Remarks on some series expansions associated with certain products ofthe incomplete Gamma functions. Comput. Math. Appl. 46:1749–1759, 2003. Submitted by.

H. M. Srivastava, 2. May 2010.

[846] H. M. Srivastava. Some bounding inequalities for the Jacobi and related functions. BanachJ. Math. Anal 1:131–138, 2007. Submitted by. H. M. Srivastava, 2. May 2010.

[847] H. M. Srivastava, R. G. Buschman. Convolution Integral Equations with Special Function

Kernels. Halsted Press (Wiley Eastern Limited, New Delhi), John Wiley and Sons, NewYork, Chichester, Brisbane and Toronto, 1977. Submitted by. H. M. Srivastava, 2. May

2010.

[848] H. M. Srivastava, R. G. Buschman. Theory and Applications of Convolution Integral Equa-tions, volume 79 of Kluwer Series on Mathematics and Its Applications. Kluwer Academic

Publishers, Dordrecht, Boston and London, 1992. Submitted by. H. M. Srivastava, 2. May

2010.[849] H. M. Srivastava, K. C. Gupta, S. P. Goyal. The H-Functions of One and Two Variables

with Applications. South Asian Publishers, New Delhi and Madras, 1982. Submitted by. H.M. Srivastava, 2. May 2010.

[850] H. M. Srivastava, P. W. Karlsson. Multiple Gaussian Hypergeometric Series. Halsted Press

(Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbaneand Toronto, 1985. Submitted by. H. M. Srivastava, 2. May 2010.

[851] H. M. Srivastava, B. R. K. Kashyap. Special Functions in Queing Theory and Related Sto-

chastic Processes. Academic Press, New York, London and San Francisco, 1982. Submittedby. H. M. Srivastava, 2. May 2010.

[852] H. M. Srivastava, H. L. Manocha. A Treatise on Generating Functions. Halsted Press (EllisHorwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane andToronto, 1984. Submitted by. H. M. Srivastava, 2. May 2010.

[853] H. M. Srivastava, S. Nadarajah, S. Kotz. Some generalizations of the Laplace distribution.

Appl. Math. Comput. 182:223–231, 2006. Submitted by. H. M. Srivastava, 2. May 2010.[854] H. M. Srivastava, A. W. Niukkanen. Some Clebsch-Gordan type linearization relations and

associated families of Dirichlet integrals. Math. Comput. Modelling 37:245–250, 2003. Sub-mitted by. H. M. Srivastava, 2. May 2010.

[855] H. M. Srivastava, S. Owa. Univalent Functions, Fractional Calculus, and Their Applica-

tions. Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York,Chichester, Brisbane and Toronto, 1989. Submitted by. H. M. Srivastava, 2. May 2010.


[856] H. M. Srivastava, M. I. Qureshi, R. Singh, A. Arora. A family of hypergeometric integralsassociated with Ramanujan’s integral formula. Adv. Stud. Contemp. Math. 18:113–125,

2009. Submitted by. H. M. Srivastava, 2. May 2010.

[857] H. M. Srivastava, R. K. Saxena. Operators of fractional integration and their applications.Appl. Math. Comput. 118:1–52, 2001. Submitted by. H. M. Srivastava, 2. May 2010.

[858] H. M. Srivastava, R. K. Saxena, C. Ram. A unified presentation of the Gamma-type func-

tions occurring in diffraction theory and associated probability distributions. Appl. Math.Comput. 162:921–947, 2005. Submitted by. H. M. Srivastava, 2. May 2010.

[859] H. M. Srivastava, F. Tasdelen, B. Sekeroglu. Some families of generating functions for theq-Konhauser polynomials. Taiwanese J. Math. 12:841–850, 2008. Submitted by. H. M. Sri-

vastava, 2. May 2010.

[860] H. M. Srivastava, Z. Tomovski. Some problems and solutions involving Mathieu’s series andits generalizations. J. Inequal. Pure Appl. Math. 5(2: Article 45):1–13, 2004. Submitted by.

H. M. Srivastava, 2. May 2010, (electronic).

[861] H.M. Srivastava. Remarks on a sum containing factorials. J. Comput. Appl. Math. 142:441–444, 2002.

[862] H.M. Srivastava, J. Choi. Series associated with the Zeta and related functions. Kluwer

Acad. Publ., 2001.[863] Ernst Stadlober. Generating Student’s t variates by a modified rejection method. In Collec-

tion: Probability and statistical inference (Bad Tatzmannsdorf, 1981), 349–360, Dordrecht–

Boston, Mass, 1982. Reidel.[864] D.D. Stancu, M.R. Occorsio. Mean-value formulae for integrals obtained by using Gaussian-

type quadratures. Rend. Circ. Mat. Palermo, Serie II(33):463–478, 1993.

[865] J. Stankovic. Uber einige Relationen zwischen Fakultaten und den linken Fakultaten. Math.

Balkanica 3:488–495, 1973.

[866] M.A. Stern. Zur Theorie des Eulerschen Integrale. Gottinger Studien, 1847.[867] Stephen M. Stigler. The history of statistics. Harvard University Press, 1986, xviii+410 pp,

$25.00, Cambridge, Mass, 1986. The measurement of uncertainty before 1900.

[868] Marius I. Stoka. La distribution normale generalisee et quelques problemes d’inference statis-tique. Rend. Sem. Mat. Univ. Politec. Torino [Rendiconti del Seminario Matematico (gia

“Conferenze di Fisica e di Matematica”) Universita e Politecnico di Torino] 37(2):157–171,

1979. French.[869] Kenneth B. Stolarsky. Gauss’ Gamma multiplication theorem: Analogues and extensions.

In Collection: The mathematical heritage of C. F. Gauss, 733–757. World Sci. Publishing,River Edge, NJ, 1991.

[870] Nariaki Sugiura, Hiromi Sasamoto. Locally best invariant test for outliers in a Gamma type

distribution. Comm. Statist. Simulation Comput. [Communications in Statistics. Simula-tion and Computation] 18(2):415–427, 1989.

[871] F.J. Sullivan. A note on the q-Gamma functions. Riv. Mat. Univ. Parma 13(4):401–405,1987.

[872] Francis J. Sullivan. A note on the q-Gamma functions. Riv. Mat. Univ. Parma (4) [Rivista

di Matematica della Universita di Parma. Serie IV] 13:401–405, 1987.

[873] J.S. Sun. Monotonicity results and inequalities involving the Gamma function. RGMIAResearch Report Collection 7(3), 2004.

[874] Noriyuki Suwa. A p-adic limit formula for Gauss sums. J. Number Theory [Journal of

Number Theory] 34(3):276–283, 1990.[875] T. Szantai. On negatively correlated Gamma random variates. Z. Angew. Math.

Mech. [Zeitschrift fur Angewandte Mathematik und Mechanik. IngenieurwissenschaftlicheForschungsarbeiten] 62(5):386–387, 1982. Proceedings of the Annual Meeting of the

Gesellschaft fur Angewandte Mathematik und Mechanik, Wurzburg 1981, Part II.

[876] T. Szantai. Evaluation of a special multivariate Gamma distribution function. Math. Pro-gramming Stud. [Mathematical Programming Study](27):1–16, 1986. Stochastic program-

ming 84. I.

[877] Tamas Szantai. An efficient algorithm for fitting a multivariate Gamma distribution to em-pirical data. Alkalmaz. Mat. Lapok [Alkalmazott Matematikai Lapok] 10(1–2):35–60, 1984.

Hungarian.


[878] Aleksander Szczuka, Ryszard Zielinski. Pseudorandom number generators with a Gammadistribution. Mat. Stos. [Roczniki Polskiego Towarzystwa Matematycznego. Seria III.

Matematyka Stosowana. Applied Mathematics] 36:99–114, 1993. Polish.

[879] P. R. Tadikamalla. Random sampling from the generalized Gamma distribution. Computing.Archiv fur Informatik und Numerik] 23(2):199–203, 1979.

[880] Tokio Taguchi. On an interpretation and an estimation of shape parameters of the general-

ized Gamma distribution. Metron 38(3–4):27–40, 1980.[881] M. Tamaki. Optimal selection from a Gamma distribution with unknown parameter. Z.

Oper. Res. Ser. A-B [Zeitschrift fur Operations Research. Serie A. Serie B] 28(1):47–57,1984.

[882] J. Tanney. Sur les integrales Euleriennes. Comptes Rendues 94:1698–1701, 1881.

[883] M. Luisa Targhetta. On a family of indeterminate distributions. J. Math. Anal. Appl. [Jour-nal of Mathematical Analysis and Applications] 147(2):477–479, 1990.

[884] N. M. Temme. The uniform asymptotic expansion of a class of integrals related to cumulative

distribution functions. SIAM J. Math. Anal. [SIAM Journal on Mathematical Analysis]13(2):239–253, 1982.

[885] N. M. Temme. The numerical computation of the confluent hypergeometric function

u(a, b, z). Numer. Math. [Numerische Mathematik] 41(1):63–82, 1983.[886] N. M. Temme. Incomplete Laplace integrals: Uniform asymptotic expansion with application

to the incomplete Beta function. SIAM J. Math. Anal. [SIAM Journal on Mathematical

Analysis] 18(6):1638–1663, 1987.[887] N. M. Temme. On the computation of the incomplete Gamma functions for large values of the

parameters. In Collection: Algorithms for approximation (Shrivenham, 1985), volume 10

of Inst. Math. Appl. Conf. Ser. New Ser., 479–489. Oxford Univ. Press, New York, 1987.[888] N. M. Temme, A. B. Olde Daalhuis. Uniform asymptotic approximation of Fermi-Dirac

integrals. J. Comput. Appl. Math. [Journal of Computational and Applied Mathematics]31(3):383–387, 1990.

[889] N.M. Temme. The asymptotic expansions of the incomplete Gamma functions. SIAM J.

Math. Anal. 10:757–766, 1979.[890] N.M. Temme. Asymptotic inversion of incomplete Gamma functions. Math. Comp.

58(198):755–764, 1992.

[891] D. Thakur. Gamma functions for function fields and Drinfeld modules. Ann Math. 134:25–64, 1991.

[892] D. S. Thakur. On Gamma functions for function fields. In Collection: The arithmetic of

function fields (Columbus, OH, 1991), volume 2 of Ohio State Univ. Math. Res. Inst. Publ,75–86. de Gruyter, Berlin, 1992.

[893] Dinesh S. Thakur. Gauss sums for fq [t]. Invent. Math. [Inventiones Mathematicae]

94(1):105–112, 1988.[894] Dinesh S. Thakur. Gross-Koblitz formula for function fields. In Collection: p-adic analysis

(Trento, 1989), volume 1454 of Lecture Notes in Math, 351–355, Berlin, 1990. Springer.[895] Dinesh S. Thakur. Gamma functions for function fields and Drinfeld modules. Ann. of Math.

(2) [Annals of Mathematics. Second Series] 134(1):25–64, 1991.

[896] Dinesh S. Thakur. Shtukas and Jacobi Sums. Invent. Math. [Inventiones Mathematicae]111(3):557–570, 1993.

[897] D.S. Thakur. On Gamma functions for function fields. In The arithmetic of function fields

(Columbus, OH, 1991), 75–86. de Gruyter, Berlin, 1992.[898] D.S. Thakur. Transcendence of Gamma values for fq [t]. Annals of Math., (2) 144(1):181–

188, 1996.[899] H.P. Thielman. On a pair of functional equations. Amer. Math. Monthly 57(8):544–547,

1950.

[900] H.P. Thielman. A note on a functional equation. Amer. J. Math 73:482–484, 1951.

[901] Alain Thiery. Transcendance de quelques valeurs de la fonction Gamma pour les corps defonctions. [Transcendence results for some values of the Gamma function for function fields].

C. R. Acad. Sci. Paris Ser. I Math. [Comptes Rendus de l’Academie des Sciences. Serie I.Mathematique] 314(13):973–976, 1992. French.

[902] P. Yageen Thomas. On some identities involving the moments of extremes from Gamma

distribution. Comm. Statist. Theory Methods [Communications in Statistics. Theory andMethods] 22(8):2321–2326, 1993.


[903] P. Yageen Thomas, T. S. K. Moothathu. Recurrence relations for moments of different ordersof extremes from Gamma distribution. Comm. Statist. Theory Methods [Communications

in Statistics. Theory and Methods] 20(3):945–950, 1991.

[904] S.R. Tims, J.A. Tyrrell. Approximate evaluation of Euler’s constant. Math. Gaz. 55:65–67,1971.

[905] E.C. Titchmarsh. The theory of the Riemann Zeta function. Oxford, 1951. (2nd ed. Revised

by D.R. Heath-Brown, 1986).[906] Eric S. Tollar. A characterization of the Gamma distribution from a random difference

equation. J. Appl. Probab. [Journal of Applied Probability] 25(1):142–149, 1988.[907] Adele Tortorici Macaluso. Generalization of some inequalities for the Gamma function. Riv.

Mat. Univ. Parma (4) [Rivista di Matematica della Universita di Parma. Serie IV] 13:373–

378, 1987. Italian.[908] L. Toth. On Wallis’ formula (Hungarian). Mat. Lapok (Cluj) 98 (41)(4):128–130, 1993. In

Hungarian.

[909] R. J. A. Tough. The estimators of normalized moments of the Gamma distribution. J. Phys.A [Journal of Physics. A. Mathematical and General] 27(23):7883–7892, 1994.

[910] Transactions of the 10th Prague Conference on Information Theory, Statistical Decision

Functions, Random Processes. Vol. A., Dordrecht, 1988. D. Reidel Publishing Co, 438 pp.Conference: Information Theory, Statistical Decision Functions, Random Processes, Held in

Prague, July 7–11, 1986.

[911] S.Y. Trimble, J. Wells, F.T. Wright. Superadditive functions and a statistical application.SIAM J. Math. Anal. 20:1255–1259, 1989.

[912] H. Tsumura. On p-adic log-γ-functions associated to the Lubin-Tate formal groups. Tokyo

J. Math. 19:147–153, 1996.[913] Hirofumi Tsumura. On a q-analogue of the log-γ-function. Nagoya Math. J. [Nagoya Math-

ematical Journal] 134:57–64, 1994.[914] Masao Tsuzuki. Elliptic factors of Selberg Zeta functions. Proc. Japan Acad. Ser. A Math.

Sci. [Japan Academy. Proceedings. Series A. Mathematical Sciences] 69(10):417–421, 1993.

[915] S. T. Tu, P. Y. Wang, H. M. Srivastava. Some families of infinite series summable viafractional calculus operators. East Asian Math. J. 18:111–125, 2002. Submitted by. H. M.


[916] S. T. Tulyaganov. A conjecture on sums of multiplicative functions. Mat. Sb. [Matematich-eskii Sbornik] 181(11):1543–1557, 1990. Russian.

[917] K. Ueno, M. Nishizawa. Multiple Gamma functions and multiple q-Gamma functions. Publ.

Res. Inst. Math. Sci. 33(5):813–838, 1997.

[918] D. Ugrin Sparac. Problem of uniqueness in the renewal process generated by the uniform dis-

tribution. J. Appl. Math. Stochastic Anal. [Journal of Applied Mathematics and StochasticAnalysis] 5(4):291–305, 1992.

[919] V. Uppuluri, R. Rao. A stronger version of Gautschi’s inequality satisfied by the Gamma

function. Skand. Aktuarietidskr.:51–52, 1965.[920] J. van de Lune. Some Convexity Properties of Euler’s Gamma function., volume 105 of

Afdeling Zuivere Wiskunde [Department of Pure Mathematics]. Mathematisch Centrum, 13

pp, Dfl. 3.00, Amsterdam, 1978.[921] J. Van de Lune. Some convexity properties of Euler’s Gamma function. Math. Centrum,

Amsterdam, 1978. 13 pp.[922] J. van de Lune, M. Voorhoeve. On the Values of a function related to Euler’s Gamma

function., volume 170 of Afdeling Zuivere Wiskunde [Department of Pure Mathematics].

Mathematisch Centrum, i+9 pp, Amsterdam, 1981.[923] C. G. van der Laan, N. M. Temme. Calculation of Special Functions: The Gamma function,

the Exponential integrals and Error-like functions., volume 10 of CWI Tract. Stichting

Mathematisch Centrum, Centrum voor Wiskunde en Informatica iv+231 pp., 1984.[924] Lucien van Hamme. The p-adic Gamma function and the congruences of Atkin and

Swinnerton-Dyer. In Collection: Study group on ultrametric analysis, 9th year: 1981/82,

No. 3 (Marseille, 1982), Exp. No. J17, 6 pp. Inst. Henri Poincare, Paris, 1983.[925] Lucien Van Hamme. Continuous operators which commute with translations, on the space

of continuous functions on zp. In Collection: p-adic functional analysis (Laredo, 1990),

volume 137 of Lecture Notes in Pure and Appl. Math, 75–88, New York, 1992. Dekker.


[926] Ilan Vardi. Determinants of Laplacians and multiple Gamma functions. SIAM J. Math.Anal. [SIAM Journal on Mathematical Analysis] 19(2):493–507, 1988.

[927] A. Vernescu. Sur l’approximation et le developpement asymptotique de la serie de terme

general(2n− 1)!!

(2n)!!. In Proc. Annual Meeting Romanian Soc. Math. Sci. Buch., volume 1,

205–213, 1977.[928] A. Vernescu. On the asymptotic analysis of the functions of discrete variable. Bul. St. Univ.

Baia Mare, Seria B, Mat.-Inf. 17(1–2):173–185, 2001.

[929] A. Vernescu. The natural proof of the inequalities of Wallis type. Libertas Mathematica24:183–190, 2004.

[930] L. Vietoris. Dritter Beweis der die Unvollstandige Gammafunktion betreffenden Lochsschen

Ungleichungen. [Third proof of Lochs’s inequalities concerning]. Osterreich. Akad. Wiss.Math. Natur. Kl. Sitzungsber. II [Osterreichische Akademie der Wissenschaften. Mathe-

matisch Naturwissenschaftliche], 1983. German.

[931] Marie France Vigneras. L’equation fonctionnelle de la fonction Zeta de Selberg du groupemodulaire psl(2, z). In Collection: Journees Arithmetiques de Luminy (Colloq. Internat.

CNRS, Centre Univ. Luminy, Luminy, 1978), volume 61 of Asterisque, 235–249. Soc. Math.France, Paris, 1979. French.

[932] N. Ja. Vilenkin, A. U. Klimyk. Representation of Lie Groups and Special Functions. Vol. 3,

volume 75 of Mathematics and its Applications (Soviet Series). Kluwer Academic PublishersGroup, xx+634 pp, $292.00, Dordrecht, 1992. Classical and quantum groups and special

functions. Translated from the Russian by V. A. Groza and A. A. Groza.

[933] V. S. Vladimirov. Freund-Witten adelic formulas for Veneziano amplitudes. Lett. Math.Phys. [Letters in Mathematical Physics] 27(2):123–131, 1993.

[934] V. S. Vladimirov. Freund-Witten adelic formulas for Veneziano and Virasoro-Shapiro am-

plitudes. Uspekhi Mat. Nauk [Rossiiskaya Akademiya Nauk. Moskovskoe MatematicheskoeObshchestvo. Uspekhi Matematicheskikh Nauk] 48(6(294)):3–38, 1993. In Russian.

[935] V. S. Vladimirov. On a justification of the Freund-Witten Adelic formula for four-point

Veneziano amplitudes. Teoret. Mat. Fiz. [Rossiiskaya Akademiya Nauk. Teoreticheskaya iMatematicheskaya Fizika] 94(3):355–367, 1993.

[936] V. S. Vladimirov. On a justification of the Freund-Witten Adelic formula for four-point

Veneziano amplitudes. Theoret. and Math. Phys. [Theoretical and Mathematical Physics]94(3):251–259, 1993. Translation.

[937] V. S. Vladimirov. On the Freund-Witten adelic formula for Veneziano amplitudes. Lett.Math. Phys. [Letters in Mathematical Physics. A Journal for the Rapid Dissemination of

Short Contributions in the Field of Mathematical Physics] 27(2):123–131, 1993.

[938] V. S. Vladimirov. Adelic formulas for four-point Veneziano and Virasoro-Shapiro amplitudes.Phys. Dokl. [Physics. Doklady] 38(12):486–489, 1994. Translation from Russian.

[939] H. Vogt, J. Voigt. A monotonocity property of the γ-function. J. Ineq. Pure Appl. Math.

3(5), 2002.[940] Andre Voros. Probleme spectral de Sturm-Liouville: le cas de l’oscillateur quartique. [The

Sturm-Liouville spectral problem: case of a quartic oscillator]. In Collection: Bourbaki

seminar, Vol. 1982/83, Asterisque, 105-106, 95–104. Soc. Math. France, Paris, 1983. French.[941] A.D. Wadhwa. An interesting subseries of the harmonic series. Amer. Math. Monthly

82:931–933, 1975.

[942] S. D. Walter, L. W. Stitt. Extended tables for moments of Gamma distribution order sta-tistics. Comm. Statist. Simulation Comput. [Communications in Statistics. Simulation and

Computation] 17(2):471–487, 1988.[943] W. Walter, editor. General inequalities 4, volume 71 of Internationale Schriftenreihe zur

Sumerischen Mathematik [International Series of Numerical Mathematics]. Birkhauser Ver-lag xxi+434 pp, $39.95, Basel-Boston, Mass., 1984. In Memorium Edwin F. Beckenbach.

[944] Guido Walz. A note on the approximation of the incomplete Gamma function by a methodof G. Meinardus. Appl. Math. Lett. [Applied Mathematics Letters. An International Journal

of Rapid Publication] 1(1):91–94, 1988.[945] Yao Hung Wang. Extensions of Lukacs’ characterization of the Gamma distribution. In

Collection: Analytical methods in probability theory (Oberwolfach, 1980), volume 861 ofLecture Notes in Math, 166–177, Berlin–New York, 1981. Springer.


[946] Z. X. Wang, Dun Ren Guo. Special Functions. World Scientific Publishing Co. Inc, xviii+695pp, $75.00., Teaneck, NJ, 1989. Translated from the Chinese by Guo and X. J. Xia.

[947] Antoni Wawrzynczyk. Wspolczesna Teoria Funkcji Specjalnych. [Contemporary Theories of

Special Functions], volume 52 of Biblioteka Matematyczna [Mathematics Library]. Panst-wowe Wydawnictwo Naukowe (PWN),, 1978, 527 pp. (errata insert), Zl 90.00, Warsaw, 1978.

Wstep do analizy harmonicznej na przestrzeniach jednorodnych. [Introduction to harmonic

analysis in homogeneous spaces] With a section on problems by Aleksander Strasburger.,Polish.

[948] James Weber. Generalized transformations of random variables. Statist. Probab. Lett. [Sta-tistics and Probability Letters] 12(2):161–166, 1991.

[949] R. Webster. Log-convex solutions to the functional equation f(x + 1) = g(x)f(x): γ-type

functions. J. Math. Anal. Appl. 209:605–623, 1997.[950] J. Weijian, G. Mingzhe, G. Kuemei. On an extension of the Jardy-Hilbert theorem. Studia

Sci. Math. Hung. 42(1):21–35, 2005.

[951] E.W. Weisstein. CRC Concise Encyclopedia of Mathematics. Boca Raton, Washington D.C.,1998. (see also www.mathworld.wolfram.com).

[952] Z.Y. Wen, J.Y. Yao. Transcendence, automate theory and Gamma functions for polynomial

rings. Acta Arith. 101(1):39–51, 2002.[953] J.G. Wendel. Note on the Gamma function. Amer. Math. Monthly 55:553–564, 1948.

[954] J. Wesolowski. A characterization of the Gamma process by conditional moments. Metrika

36(5):299–309, 1989.[955] Jacek Wesolowski. A constant regression characterization of the Gamma law. Adv. in Appl.

Probab. [Advances in Applied Probability] 22(2):488–490, 1990.

[956] E.T. Whittaker, G.N. Watson. A Course of Modern Analysis. Cambridge University Press,4th edition, 1962. part II, Chapter 12.

[957] H.S. Wilf. The asymptotic behaviour of the Stirling numbers of the first kind. J. Combin.Theory Ser. A 64:344–349, 1993.

[958] R. William Gosper, M.E.H. Ismail, Ruiming Zhang. On some strange summation formulas.

Illinois J. Math. 37(2):240–277, 1993.[959] K.P. Williams. The asymptotic form of the function ψ(x). Bull. Amer. M.S. 19:472–479,

1913.

[960] K.G. Wilson. Proof of a conjecture of Dyson. J. Math. Phys. 3:1040–1043, 1962.[961] C. S. Withers. Tests and confidence intervals for the shape parameter of a Gamma distribu-

tion. Comm. Statist. Theory Methods [Communications in Statistics. Theory and Methods]

20(8):2663–2685, 1991.[962] C. S. Withers. Corrigendum: “Tests and confidence intervals for the shape parame-

ter of a Gamma distribution” [Comm. Statist. Theory Methods 20, no 8 (1991), 2663–

2685]. Comm.-Statist.-Theory-Methods [Communications-in-Statistics.-Theory and Meth-ods] 21(12):3625–3626, 1992.

[963] W. Wolfe. The asymptotic distribution of lattice points in hyperbolic space. J. Funct. Anal-ysis 31:333–340, 1979.

[964] Augustine C. M. Wong. A note on inference for the mean parameter of the Gamma distri-

bution. Statist. Probab. Lett. [Statistics and Probability Letters] 17(1):61–66, 1993.[965] J.W. Wrench Jr. Concerning two series for the Gamma function. Math. Comp. 22:617–626,

1968.

[966] Staffan Wrigge. A note on the moment generating function for the reciprocal Gamma dis-tribution. Math. Comp. [Mathematics of Computation] 42(166):617–621, 1984.

[967] Randall Wright. Expectation dependence of random variables, with an application in port-folio theory. Theory and Decision [Theory and Decision. An International Journal for Phi-losophy and Methodology of the Social Sciences] 22(2):111–124, 1987.

[968] Ming Guan Wu. A fast algorithm for the incomplete Gamma function. Heilongjiang

Shuizhuan Xuebao [Heilongjiang Shuizhuan Xuebao. Journal of Heilongjiang Hydraulic En-gineering College] 18(4):57–62, 1991. In Chinese.

[969] T. C. Wu, S. H. Leu, S. T. Tu, H. M. Srivastava. A certain class of infinite sums associ-ated with Digamma functions. Appl. Math. Comput. 105:1–9, 1999. Submitted by. H. M.



[970] T. C. Wu, S. T. Tu, H. M. Srivastava. Some combinatorial series identities associated withthe Digamma function and harmonic numbers. Appl. Math. Lett. 13(3):101–106, 2000. Sub-

mitted by. H. M. Srivastava, 2. May 2010.

[971] Guijing Xiong. On a kind of the best estimates for the Euler constant γ. Acta Math. Scientia:English Ed. 16:458–468, 1996.

[972] T. Yanagimoto. The conditional maximum likelihood estimator of the shape parameter in

the Gamma distribution. Metrika 35(3–4):161–175, 1988.[973] S. Yang, H. M. Srivastava. Some generalizations of the Hadamard expansion for the modified

Bessel function. Appl. Math. Lett. 17:591–596, 2004. Submitted by. H. M. Srivastava, 2. May2010.

[974] Nikos Yannaros. On Cox processes and Gamma renewal processes. J. Appl. Probab. [Journal

of Applied Probability] 25(2):423–427, 1988.[975] A. Y. Yehia. Characterization of a mixture of generalized Gamma distribution. Egyptian

Statist. J. [The Egyptian Statistical Journal] 36(1):95–101, 1992.

[976] Kin Wah Yu. On the central-limit theorem of an infinite sum of independent Pareto-Levyrandom variables. Phys. Lett. A [Physics Letters. A] 116(3):145–147, 1986.

[977] Ahmed I. Zayed. A proof of new summation formulae by using sampling theorems. Proc.

Amer. Math. Soc. [Proceedings of the American Mathematical Society] 117(3):699–710,1993.

[978] Doron Zeilberger. Sister Celine’s technique and its generalizations. J. Math. Anal. Appl.

[Journal of Mathematical Analysis and Applications] 85(1):114–145, 1982.[979] Changgui Zhang. La fonction Gamma consideree comme la somme de nombres Euleriens

generalises. [The Gamma function considered as the sum of generalized Eulerian numbers].

European J. Combin. [European Journal of Combinatorics] 15(3):309–315, 1994.[980] You Cheng Zhou. Continua and the set function γ. II. Chinese Ann. Math. Ser. A [Chinese

Annals of Mathematics. Series A. Shuxue Niankan. Ji A] 11(3):339–343, 1990. Chinese.[981] You Cheng Zhou. Continuous mappings and the set function γ. J. Math. (Wuhan) [Journal

of Mathematics. Shuxue Zazhi] 10(4):469–472, 1990. Chinese.

[982] Chen Bo Zhu. Functional equations satisfied by intertwining operators of reductive groups.Trans. Amer. Math. Soc. [Transactions of the American Mathematical Society] 336(2):881–

899, 1993.

[983] V. A. Zmorovich, I. K. Korobkova. On a class of Gammamorphic functions. Ukrainian Math.J. [Ukrainian Mathematical Journal] 41(11):1347–1348, 1990. Translation from Ukrain.

Mat. Zh. [Akademiya Nauk Ukrainskoi SSR. Institut Matematiki. Ukrainskii Matematich-

eskii Zhurnal], 41, No 11, 1989.[984] A. F. Zubova. Calculation of the reliability of machines and apparatus in the case of Gamma

distribution. In Collection: Some problems on the qualitative theory of differential equations

and the theory of the control of motion, III, 51–55. Mordov. Gos. Univ, Saransk, 1983.Russian.

[985] I.J. Zucker. The evaluation in terms of γ-functions of the periods of elliptic curves admittingcomplex multiplication. Proc. Camb. Phil. Soc. 82:111–118, 1977.

[986] V.V. Zudilin. Remarks on irrationality of q-harmonic series. Manuscripta Math. 107(4):463–

477, 2002.

a bibliography of gamma function and related … · matematika](6):3{8, 90, 1987. russian. [6] c....

Documents