special hypergeometric functions q

Special hypergeometric functions qFrom Wikipedia, the free encyclopedia

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

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

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

1

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

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





2.7 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



3

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




3.6 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.


4

3.7. REFERENCES 5



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







6

4.7. REFERENCES 7


Stanton, Dennis (1981), Three addition theorems for some q-Krawtchouk polynomials, Geometriae Dedicata10 (1): 403425, doi:10.1007/BF01447435, ISSN 0046-5755, MR 608153

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





5.7 References



8

5.7. REFERENCES 9


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

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.


6.2 Orthogonality







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

6.7. REFERENCES 11

F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248

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





7.7 References[1] Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analoques, p460,Springer



12

7.7. REFERENCES 13


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.


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




8.6 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

8.7. REFERENCES 15




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.


9.2 Orthogonality








16

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

special hypergeometric functions q

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qcharlier polynomials

qhahn polynomials

qlaguerre polynomials

qmeixner polynomials

qmeixnerpollaczek polynomials

qracah polynomials

denitionthe polynomials

qbessel polynomials