special hypergeometric functions q
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Special hypergeometric functions qFrom Wikipedia, the free encyclopedia
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Contents
1 q-Bessel polynomials 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Recurrence and dierence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Relation to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Gallery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 q-Charlier polynomials 32.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Recurrence and dierence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 Relation to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 q-Hahn polynomials 43.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Recurrence and dierence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.5 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.6 Relation to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 q-Krawtchouk polynomials 64.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.3 Recurrence and dierence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.4 Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
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ii CONTENTS
4.5 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.6 Relation to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 q-Laguerre polynomials 85.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.3 Recurrence and dierence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.4 Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.5 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.6 Relation to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6 q-Meixner polynomials 106.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.3 Recurrence and dierence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.4 Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.5 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.6 Relation to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7 q-MeixnerPollaczek polynomials 127.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.3 Recurrence and dierence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.4 Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.5 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.6 Relation to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
8 q-Racah polynomials 148.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.3 Recurrence and dierence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.4 Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.5 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.6 Relation to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
9 Quantum q-Krawtchouk polynomials 169.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
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CONTENTS iii
9.3 Recurrence and dierence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.4 Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.5 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.6 Relation to other polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 17
9.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
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Chapter 1
q-Bessel polynomials
In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basicAskey scheme. Roelof Koekoek, Peter A. Lesky, and Ren F. Swarttouw (2010, 14) give a detailed list of theirproperties.
1.1 DenitionThe polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by [1]:
yn(x; a; q) = 21
qN aqn0
; q; qx
1.2 OrthogonalityP1
k=0(ak
(q;q)n q(k+12 ) ym (qk; a; q) yn (qk; a; q) = (q; q)n (aqn; q)1 a
nq(n+12 )
1+aq2n mn[2]
1.3 Recurrence and dierence relations
1.4 Rodrigues formula
1.5 Generating function
1.6 Relation to other polynomials
1.7 Gallery
1.8 References[1] Roelof Koekoek, Peter Lesky Rene Swarttouw,Hypergeometric Orthogonal Polynomials and their q-Analogues, p526
Springer 2010
[2] Roelof p527
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2 CHAPTER 1. Q-BESSEL POLYNOMIALS
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldF.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
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Chapter 2
q-Charlier polynomials
In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in thebasic Askey scheme. Roelof Koekoek, Peter A. Lesky, and Ren F. Swarttouw (2010, 14) give a detailed list of theirproperties.
2.1 DenitionThe q-Charlier polynomials are given in terms of the basic hypergeometric function by
cn(qx; a; q) = 21(qn; qx; 0; q;qn+1/a)
2.2 Orthogonality
2.3 Recurrence and dierence relations
2.4 Rodrigues formula
2.5 Generating function
2.6 Relation to other polynomials
2.7 References Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldF.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
3
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Chapter 3
q-Hahn polynomials
See also: continuous q-Hahn polynomials, dual q-Hahn polynomials and continuous dual q-Hahn polynomials
In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basicAskey scheme. Roelof Koekoek, Peter A. Lesky, and Ren F. Swarttouw (2010, 14) give a detailed list of theirproperties.
3.1 DenitionThe polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol byQn(x; a; b;N ; q) =32
qn abqn + 1 xaq qN ; q; q
3.2 Orthogonality
3.3 Recurrence and dierence relations
3.4 Rodrigues formula
3.5 Generating function
3.6 Relation to other polynomialsq-Hahn polynomials Quantum q-Krawtchouk polynomials:lima!1Qn(qx; a; p;N jq) = Kqtmn (qx; p;N ; q)q-Hahn polynomials Hahn polynomialsmake the substitution = q , = q into denition of q-Hahn polynomials, and nd the limit q1, we obtain: 3F2([n; + + n+ 1;x]; [+ 1;N ]; 1) ,which is exactly Hahn polynomials.
3.7 References Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719
4
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3.7. REFERENCES 5
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldF.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
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Chapter 4
q-Krawtchouk polynomials
See also: ane q-Krawtchouk polynomials, dual q-Krawtchouk polynomials and quantum q-Krawtchouk polynomi-als
In mathematics, the q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in thebasic Askey scheme. Roelof Koekoek, Peter A. Lesky, and Ren F. Swarttouw (2010, 14) give a detailed list of theirproperties.Stanton (1981) showed that the q-Krawtchouk polynomials are spherical functions for 3 dierent Chevalley groupsover nite elds, and Koornwinder (1989) showed that they are related to representations of the quantum group SU(2).
4.1 DenitionThe polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by
4.2 Orthogonality
4.3 Recurrence and dierence relations
4.4 Rodrigues formula
4.5 Generating function
4.6 Relation to other polynomials
4.7 References Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
6
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4.7. REFERENCES 7
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldF.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
Stanton, Dennis (1981), Three addition theorems for some q-Krawtchouk polynomials, Geometriae Dedicata10 (1): 403425, doi:10.1007/BF01447435, ISSN 0046-5755, MR 608153
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Chapter 5
q-Laguerre polynomials
See also: big q-Laguerre polynomials, continuous q-Laguerre polynomials and little q-Laguerre polynomials
In mathematics, the q-Laguerre polynomials, or generalized StieltjesWigert polynomials P()n(x;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by DanielS. Moak (1981). Roelof Koekoek, Peter A. Lesky, and Ren F. Swarttouw (2010, 14) give a detailed list of theirproperties.
5.1 DenitionThe q-Laguerre polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by
L()n (x; q) =(q+1; q)n(q; q)n
11(qn; q+1; q;qn++1x)
5.2 Orthogonality
5.3 Recurrence and dierence relations
5.4 Rodrigues formula
5.5 Generating function
5.6 Relation to other polynomials
5.7 References
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
8
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5.7. REFERENCES 9
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldF.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
Moak, Daniel S. (1981), The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1): 2047,doi:10.1016/0022-247X(81)90048-2, MR 618759
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Chapter 6
q-Meixner polynomials
Not to be confused with q-MeixnerPollaczek polynomials.
In mathematics, the q-Meixner polynomials are a family of basic hypergeometric orthogonal polynomials in thebasic Askey scheme. Roelof Koekoek, Peter A. Lesky, and Ren F. Swarttouw (2010, 14) give a detailed list of theirproperties.
6.1 DenitionThe polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by
6.2 Orthogonality
6.3 Recurrence and dierence relations
6.4 Rodrigues formula
6.5 Generating function
6.6 Relation to other polynomials
6.7 References Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald
10
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6.7. REFERENCES 11
F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
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Chapter 7
q-MeixnerPollaczek polynomials
Not to be confused with q-Meixner polynomials.
Inmathematics, the q-MeixnerPollaczek polynomials are a family of basic hypergeometric orthogonal polynomialsin the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and Ren F. Swarttouw (2010, 14) give a detailed listof their properties.
7.1 DenitionThe polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by :[1]
Pn(x; ajq) = anein a2;qn
(q;q)n 32(q
n; aei(+2); aei; a2; 0jq; q)
7.2 Orthogonality
7.3 Recurrence and dierence relations
7.4 Rodrigues formula
7.5 Generating function
7.6 Relation to other polynomials
7.7 References[1] Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analoques, p460,Springer
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
12
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7.7. REFERENCES 13
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldF.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
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Chapter 8
q-Racah polynomials
In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basicAskey scheme, introduced by Askey & Wilson (1979). Roelof Koekoek, Peter A. Lesky, and Ren F. Swarttouw(2010, 14) give a detailed list of their properties.
8.1 DenitionThe polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by
pn(qx + qx+1cd; a; b; c; d; q) = 43
qn abqn+1 qx qx+1cdaq bdq cq
; q; q
They are sometimes given with changes of variables as
Wn(x; a; b; c;N ; q) = 43
qn abqn+1 qx cqxn
aq bcq qN ; q; q
8.2 Orthogonality
8.3 Recurrence and dierence relations
8.4 Rodrigues formula
8.5 Generating function
8.6 Relation to other polynomialsq-Racah polynomialsRacah polynomials
8.7 References Askey, Richard; Wilson, James (1979), A set of orthogonal polynomials that generalize the Racah coecientsor 6-j symbols, SIAM Journal on Mathematical Analysis 10 (5): 10081016, doi:10.1137/0510092, ISSN0036-1410, MR 541097
14
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8.7. REFERENCES 15
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldF.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
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Chapter 9
Quantum q-Krawtchouk polynomials
In mathematics, the quantum q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polyno-mials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and Ren F. Swarttouw (2010, 14) give a detailedlist of their properties.
9.1 DenitionThe polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by
9.2 Orthogonality
9.3 Recurrence and dierence relations
9.4 Rodrigues formula
9.5 Generating function
9.6 Relation to other polynomials
9.7 References Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and itsApplications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8,MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, Ren F. (2010), Hypergeometric orthogonal polynomials andtheir q-analogues, SpringerMonographs inMathematics, Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldF.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
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9.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 17
9.8 Text and image sources, contributors, and licenses9.8.1 Text
Q-Bessel polynomials Source: https://en.wikipedia.org/wiki/Q-Bessel_polynomials?oldid=655473534 Contributors: R.e.b., Headbomb,JL-Bot, Yobot and
Q-Charlier polynomials Source: https://en.wikipedia.org/wiki/Q-Charlier_polynomials?oldid=666261212Contributors: Gisling, R.e.b.,Headbomb and Yobot
Q-Hahn polynomials Source: https://en.wikipedia.org/wiki/Q-Hahn_polynomials?oldid=662811966Contributors: HaeB,Gisling, R.e.b.,Headbomb, JL-Bot, AnomieBOT and Anonymous: 1
Q-Krawtchouk polynomials Source: https://en.wikipedia.org/wiki/Q-Krawtchouk_polynomials?oldid=448547379 Contributors: R.e.b.and Headbomb
Q-Laguerre polynomials Source: https://en.wikipedia.org/wiki/Q-Laguerre_polynomials?oldid=662668138Contributors: Gisling, R.e.b.,Headbomb, JL-Bot and Rscosa
Q-Meixner polynomials Source: https://en.wikipedia.org/wiki/Q-Meixner_polynomials?oldid=448547542 Contributors: R.e.b., Head-bomb and JL-Bot
Q-MeixnerPollaczek polynomials Source: https://en.wikipedia.org/wiki/Q-Meixner%E2%80%93Pollaczek_polynomials?oldid=662668600Contributors: Gisling, R.e.b., Headbomb and Yobot
Q-Racah polynomials Source: https://en.wikipedia.org/wiki/Q-Racah_polynomials?oldid=662669429 Contributors: Gisling, R.e.b.,Headbomb and JL-Bot
Quantumq-Krawtchouk polynomials Source: https://en.wikipedia.org/wiki/Quantum_q-Krawtchouk_polynomials?oldid=448548243Contributors: R.e.b., Headbomb and JL-Bot
9.8.2 Images File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-
main Contributors: Own work, based o of Image:Ambox scales.svg Original artist: Dsmurat (talk contribs) File:QBessel_function_Im_complex_3D_Maple_plot.gif Source: https://upload.wikimedia.org/wikipedia/commons/4/4e/QBessel_
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q-Bessel polynomialsDefinitionOrthogonalityRecurrence and difference relationsRodrigues formulaGenerating functionRelation to other polynomialsGalleryReferences
q-Charlier polynomialsDefinitionOrthogonalityRecurrence and difference relationsRodrigues formulaGenerating functionRelation to other polynomialsReferences
q-Hahn polynomialsDefinitionOrthogonalityRecurrence and difference relationsRodrigues formulaGenerating functionRelation to other polynomialsReferences
q-Krawtchouk polynomialsDefinitionOrthogonalityRecurrence and difference relationsRodrigues formulaGenerating functionRelation to other polynomialsReferences
q-Laguerre polynomialsDefinitionOrthogonalityRecurrence and difference relationsRodrigues formulaGenerating functionRelation to other polynomialsReferences
q-Meixner polynomialsDefinitionOrthogonalityRecurrence and difference relationsRodrigues formulaGenerating functionRelation to other polynomialsReferences
q-MeixnerPollaczek polynomialsDefinitionOrthogonalityRecurrence and difference relationsRodrigues formulaGenerating functionRelation to other polynomialsReferences
q-Racah polynomialsDefinitionOrthogonalityRecurrence and difference relationsRodrigues formulaGenerating functionRelation to other polynomialsReferences
Quantum q-Krawtchouk polynomialsDefinitionOrthogonalityRecurrence and difference relationsRodrigues formulaGenerating functionRelation to other polynomialsReferencesText and image sources, contributors, and licensesTextImagesContent license