a note on expansions involving meijer's functions

8/3/2019 A Note on Expansions Involving Meijer's Functions.

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A NOTE ON EXPANSIONS INVOLVING MEIJER'S -FUNCTIONS 107

[7] has eonjectured that

In p(g) ~ y/g.

The maximum difference observed is 300, whereas in this region a gap of 1040

may exist.(e) Largest pair: The largest observed pair is

1000000000149342 1.

Acknowledgement. We wish to thank the referee for his suggestions as to the presentation of the above data.

Department of MathematicsMemorial University of NewfoundlandSt. John's, Newfoundland

1. D. H. Lehmer, "Tables concerning the distribution of primes up to 37 millions," 1957copy deposited in the UMT File & reviewed in M TAC, v. 13, 1959, pp. 56-57.

2. G. H. Hardy & J. E. Littlewood, "Some problems of 'partitio numerorum'; III: Onthe expression of a number as a sum of primes," Acta Math. v. 44, 1923, pp. 1-70.

3. C. L. Baker & F. J. Gruenberger, The First Six Million Prime Numbers, The RandCorporation, Santa Monica, published by the Microcard Foundation, Madison, Wisconsin,1959. Reviewed in Math. Comp., v. 15, 1961, p. 82, RMT4.

4. F. Gruenberger & G. Armerding, "Statistics on first six million prime numbers,"Paper P-2460 of the Rand Corporation, Santa Monica, 1961, 145 pp., 8J x 11 in. Copy depositedin UMT File & reviewed in Math. Comp., v. 19, 1965, pp. 503-505.

5. G. G. Alway, "A method of factorisation using a high-speed computer," MTAC, v. 6,no. 37, 1952,pp. 59-60.

6. D. N. Lehmer, List of Prime Numbers from 1 to 10,006,721, Hafner, New York, 1956.7. D. Shanks, "On maximal gaps between successive primes," Math. Comp., v. 18, 1964,pp. 646-650. MR 29 #4745.

A Note on Expansions InvolvingMeijer's G-Functions

By Arun Verma

1. Introduction. The expansions of Meijer's G-functions in a series of similarfunctions and their products with terminating hypergeometric functions, have

been studied by several mathematicians as for example Meijer [6], Wimp andLuke [10] and others. It has been shown by the author [7] that these expansionscan be written out easily from the known expansions of elementary functions byusing induction through Laplace transform and its inverse. However, it is strangeto notice that there is not even a single known expansion of Meijer's G-f unctionin a series of product of G-functions. Also recently, the author [8], [9] hasobtained the expansions of G-functions of two variables (defined by Agarwal [1])

in a series of similar functions and in a series of products of G-functions of twovariables and terminating hypergeometric functions. In this paper, using theLaplace transform and its inverse, expansions of Meijer's G-function and its exten-sion in a series of products of similar functions are obtained. The results given are

Received June 8, 1966.


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108 ARUN VERMA

just to illustrate the method and obviously a variety of similar results could bemultiplied.

2. Notations. Let

[a], = aia + l)(o + 2) (a + n - 1), [a]0 = 1,

then the generalised hypergeometric function rF is defined as

(1) 4*>']-i* r


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A NOTE ON EXPANSIONS INVOLVING MEIJER S G-FUNCTIONS 109

and

arg a; | < ir[mi + vi + n - (p 4- q + s + t)],

arg y | < 7r[m2 + v2 + n (p + q + s + t')].

(1)

3. Cooke [4] has shown that00

X) i\mx)~'lJliimx)i\my)~rJ,imy)ml

1 1/2+2T[y. + 1, v 4- 1] yT[u + 1, v 4- ]

X 2Fi 2 ", 2;m + l > ?/ > X > 0,m, ? > -i

In the notation of G-functions (3.1) can be rewritten as

(2)gK(^|m + ,)0K(; j = r[ ; 1 + v, 1 4- m]

"rinx


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110 ARUN VERMA

and multiply both sides of (3.3) by i. Then taking the inverse Laplace transformwith respect to "t" and using the result

f e'r* dt = -L, Rl 7 > 0,Jc T[y]

we find the relation (3.3) with p replaced by [p 4- 1]. Similarly, the inductionwith respect to s, q, R, S, P and Q can be completed. But for r = 0 = s = p =q = R = S = P = R, (3.3) reduces to (3.2) and this completes the proof of(3.3), by induction.

On the other hand, if we start from the following result due to Buchholz [2] :

Jxy,,n)Jxy,.n) . ttJxz) [Jy{z)Yr(Xz) - J Xz)Y ,iz)\,m=l [Jv+li-,.n)\2iZ2 yl.n) 4/,(z)

0 I i I A* i 1, and v is not an integer, the zeros of z~yJ,iz) being arranged inthe ascending magnitude of Rl(y,,,n) > 0, are y,,n in = 0, 1, 2, ), and we makeuse of the Laplace transform and its inverse, to get the following very generalresult :

2 rrP.+1 ( x I 1, (ar), 1 4 A np,Q+i ( x I 1, iaR), 1 4- v\ tv.n tri+r. I ~ /. \ 1 _!__ xn.n I JPs)_/l [/,+l(7,.n)]2(22 - 7.0

>.vnp.q+i(xI 1, (ar), 1 4- Ar, , r / s,= TTZGl+r., I - \A . ' 1 {C0t TTvJ.iz)

- F,(2)}G2+BlS [~2\(s) )

-cosecx.G^dl^"'^'1)],where

0 x ^ X 1, O^ r, 0 P ^S, 0 g Q ,

0 p ^ s, r + s V^ Mm+n [b]m [b )n m n

,=.0 n=0 [l]m [l]n [C]m [c']n


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A NOTE ON EXPANSIONS INVOLVING MEIJER's G-FUNCTIONS 111

Then rewriting (4.1) in terms of G-functions of two and one variables using[5, 5.6(1)] and [1, 3-iv], respectively, we get

Gu,i.i.i[l:l],0,[2:2]

(2)

c 4- c a I1 4- 6 - c, 1 4- 6'

= Z(c-t-c')t

f=i r\ T[a + r]c - 1,0; c - 1,0

v ^2,1 /l I c - r, 2c - 1 \\x | a + c 1, b 4- c 1/

G2 i/llc' - r, 2c' - 14- c - 1, 6' + c -0' x I 4- I y I < 1.

Then using the Laplace transform and its inverse, we can generalise (4.2) to

the result:

^l.l+p.l+S.l+r.l+ivJl,[l+P:l+o],0,[2+B:2+T]

(3)

c 4- c a 11 4-6 -c,(a,);l + 6' -c', (60)

'c- 1, (cs),0;c' - 1, (dr),0

^1+,.2+ / | 2 a - c', 2 - b c, idT)\

= E c-\~c')-ri7=r\T[a+r]

Gl+p,2+r

2+B.2+P

where

\x\ + \y\


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112 W. T. MOODY

6. C. S. Meijer, "Expansion theorems for the G-functions. I-X," Indag. Math., v. 14,1952,pp. 369-379and 483^87; v. 15, 1953, pp. 43^9, 187-193and 349-357; v. 16, 1954, pp. 77-82,83-91 and 273-279; v. 17, 1955, pp. 243-251 and 309-314. MR 14, 469; MR 14, 642; MR 14, 748;MR 14, 979; MR 15, 422; MR 16, 791; MR 15, 955; MR 16, 1106.

7. A. Verma, "A class of expansions of G-functions and the Laplace transform," Math.Comp., v. 19, 1965, pp. 661-664.

8. A. Verma, "Expansions of hypergeometric functions of two variables", Math. Comp.,v. 20, 1966,590-596.9. A. Verma, "A note on an expansion of hypergeometric functions of two variables",

Math. Comp., v. 20, 1966,413-417..10. J. Wimp & Y. L. Luke, "Expansion formulas for generalised hypergeometric func-

tions", Rend. Circ. Mat. Palermo, (2), v. 11, 1962, pp. 351-366.

Approximations for the Psi (Digamma) FunctionBy William T. Moody

A series of approximations has been derived for the psi function. As used here,the psi function is defined as the derivative of the natural logarithm of the gammafunction ; that is

, s _ dlln r(s)] _ T'jx)*K ' ' dx " IT '

The approximations are best in the Chebyshev sense, in that the magnitude ofthe maximum error in the prescribed interval is minimized. Each approximation isof the form

-7 "

x 2 j=iwherein

7 = 0.5772 , (Euler's constant).

Values of the constants, c,, and the limiting values of e for n = A, 5, 6 are givenin Table 1 below. The error of the approximation vanishes at the end points.

Table 1Values of Constants

0 S i 1,

n .

e < 1.3 X 10-6 1.3 X 10-7 1.3 X 10-8

c,

12345

6

+0.644876-0.2011864-0.077968-0.026867

4-0.6449266-0.20190404-0.0812656-0.03345324-0.0111653

4-0.64493313-0.202031814-0.08209433-0.035916654-0.01485925

-0.00472050

Bureau of ReclamationFederal CenterDenver, Colorado

Received March 28, 1966. Revised August 4, 1966.

a note on expansions involving meijer's functions

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