The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Roelof Koekoek Ren´eF. Swarttouw Abstract We list the so-called Askey-scheme of hypergeometric orthogonal polynomials and we give a q- analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence rela- tion, the second order differential or difference equation, the forward and backward shift operator, the Rodrigues-type formula and generating functions of all classes of orthogonal polynomials in this scheme. In chapter 2 we give the limit relations between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation, second order difference equation, forward and backward shift operator, Rodrigues-type formula and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal poly- nomials. Finally, in chapter 5 we point out how the ‘classical’ hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues. Acknowledgement We would like to thank Professor Tom H. Koornwinder who suggested us to write a report like this. He also helped us solving many problems we encountered during the research and provided us with several references. Contents Preface 5 Definitions and miscellaneous formulas 7 0.1 Introduction . 7 0.2 The q-shifted factorials . 8 0.3 The q-gamma function and the q-binomial coefficient . 10 0.4 Hypergeometric and basic hypergeometric functions . 11 0.5 The q-binomial theorem and other summation formulas . 13 0.6 Transformation formulas . 15 0.7 Some special functions and their q-analogues . 18 0.8 The q-derivative and the q-integral . 20 0.9 Shift operators and Rodrigues-type formulas . 21 ASKEY-SCHEME 23 1 Hypergeometric orthogonal polynomials 24 1.1 Wilson . 24 1.2 Racah . 26 1.3 Continuous dual Hahn . 29 1.4 Continuous Hahn . 31 1.5 Hahn . 33 1.6 Dual Hahn . 34 1.7 Meixner-Pollaczek . 37 1.8 Jacobi . 38 1.8.1 Gegenbauer / Ultraspherical . 40 1.8.2 Chebyshev . 41 1.8.3 Legendre / Spherical . 44 1.9 Meixner . 45 1.10 Krawtchouk . 46 1.11 Laguerre . 47 1.12 Charlier . 49 1.13 Hermite . 50 2 Limit relations between hypergeometric orthogonal polynomials 52 2.1 Wilson . 52 2.2 Racah . 52 2.3 Continuous dual Hahn . 53 2.4 Continuous Hahn . 54 2.5 Hahn . 54 2.6 Dual Hahn . 55 2.7 Meixner-Pollaczek . 55 2.8 Jacobi . 56 1 2.8.1 Gegenbauer / Ultraspherical . 57 2.9 Meixner . 57 2.10 Krawtchouk . 58 2.11 Laguerre . 58 2.12 Charlier . 59 2.13 Hermite . 59 q-SCHEME 61 3 Basic hypergeometric orthogonal polynomials 63 3.1 Askey-Wilson . 63 3.2 q-Racah . 66 3.3 Continuous dual q-Hahn . 68 3.4 Continuous q-Hahn . 71 3.5 Big q-Jacobi . 73 3.5.1 Big q-Legendre . 75 3.6 q-Hahn . 76 3.7 Dual q-Hahn . 78 3.8 Al-Salam-Chihara . 80 3.9 q-Meixner-Pollaczek . 82 3.10 Continuous q-Jacobi . 83 3.10.1 Continuous q-ultraspherical / Rogers . 86 3.10.2 Continuous q-Legendre . 89 3.11 Big q-Laguerre . 91 3.12 Little q-Jacobi . 92 3.12.1 Little q-Legendre . 94 3.13 q-Meixner . 95 3.14 Quantum q-Krawtchouk . 96 3.15 q-Krawtchouk . 98 3.16 Affine q-Krawtchouk . 100 3.17 Dual q-Krawtchouk . 101 3.18 Continuous big q-Hermite . 103 3.19 Continuous q-Laguerre . 105 3.20 Little q-Laguerre / Wall . 107 3.21 q-Laguerre . 108 3.22 Alternative q-Charlier . 110 3.23 q-Charlier . 112 3.24 Al-Salam-Carlitz I . 113 3.25 Al-Salam-Carlitz II . 114 3.26 Continuous q-Hermite . 115 3.27 Stieltjes-Wigert . 116 3.28 Discrete q-Hermite I . 118 3.29 Discrete q-Hermite II . 119 4 Limit relations between basic hypergeometric orthogonal polynomials 121 4.1 Askey-Wilson . 121 4.2 q-Racah . 122 4.3 Continuous dual q-Hahn . 123 4.4 Continuous q-Hahn . 124 4.5 Big q-Jacobi . 124 4.6 q-Hahn . 125 4.7 Dual q-Hahn . 126 4.8 Al-Salam-Chihara . 126 4.9 q-Meixner-Pollaczek . 127 2 4.10 Continuous q-Jacobi . 127 4.10.1 Continuous q-ultraspherical / Rogers . 128 4.11 Big q-Laguerre . ..