The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Roelof Koekoek Ren´e F. 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,thesecondorderdifferentialordifferenceequation,theforwardandbackwardshiftoperator, 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,orthogonalityrelation,threetermrecurrencerelation,secondorderdifferenceequation, 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.12 Little q-Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.13 q-Meixner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.14 Quantum q-Krawtchouk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.15 q-Krawtchouk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.16 Affine q-Krawtchouk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.17 Dual q-Krawtchouk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.18 Continuous big q-Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.19 Continuous q-Laguerre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.20 Little q-Laguerre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.21 q-Laguerre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.22 Alternative q-Charlier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.23 q-Charlier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.24 Al-Salam-Carlitz I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.25 Al-Salam-Carlitz II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.26 Continuous q-Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.27 Stieltjes-Wigert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.28 Discrete q-Hermite I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.29 Discrete q-Hermite II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5 From basic to classical hypergeometric orthogonal polynomials 138 5.1 Askey-Wilson → Wilson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.2 q-Racah → Racah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.3 Continuous dual q-Hahn → Continuous dual Hahn . . . . . . . . . . . . . . . . . . 138 5.4 Continuous q-Hahn → Continuous Hahn . . . . . . . . . . . . . . . . . . . . . . . . 138 5.5 Big q-Jacobi → Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.5.1 Big q-Legendre → Legendre / Spherical . . . . . . . . . . . . . . . . . . . . 139 5.6 q-Hahn → Hahn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.7 Dual q-Hahn → Dual Hahn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.8 Al-Salam-Chihara → Meixner-Pollaczek . . . . . . . . . . . . . . . . . . . . . . . . 139 5.9 q-Meixner-Pollaczek → Meixner-Pollaczek . . . . . . . . . . . . . . . . . . . . . . . 140 5.10 Continuous q-Jacobi → Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.10.1 Continuous q-ultraspherical / Rogers → Gegenbauer / Ultraspherical. . . . 140 5.10.2 Continuous q-Legendre → Legendre / Spherical . . . . . . . . . . . . . . . . 140 5.11 Big q-Laguerre → Laguerre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.12 Little q-Jacobi → Jacobi / Laguerre . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.12.1 Little q-Legendre → Legendre / Spherical . . . . . . . . . . . . . . . . . . . 141 5.13 q-Meixner → Meixner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.14 Quantum q-Krawtchouk → Krawtchouk . . . . . . . . . . . . . . . . . . . . . . . . 141 5.15 q-Krawtchouk → Krawtchouk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.16 Affine q-Krawtchouk → Krawtchouk . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.17 Dual q-Krawtchouk → Krawtchouk . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.18 Continuous big q-Hermite → Hermite . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.19 Continuous q-Laguerre → Laguerre . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.20 Little q-Laguerre / Wall → Laguerre / Charlier . . . . . . . . . . . . . . . . . . . . 142 5.21 q-Laguerre → Laguerre / Charlier . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.22 Alternative q-Charlier → Charlier. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.23 q-Charlier → Charlier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.24 Al-Salam-Carlitz I → Charlier / Hermite . . . . . . . . . . . . . . . . . . . . . . . 143 5.25 Al-Salam-Carlitz II → Charlier / Hermite . . . . . . . . . . . . . . . . . . . . . . . 143 5.26 Continuous q-Hermite → Hermite. . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.27 Stieltjes-Wigert → Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3 5.28 Discrete q-Hermite I → Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.29 Discrete q-Hermite II → Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Bibliography 145 Index 167 4 Preface This report deals with orthogonal polynomials appearing in the so-called Askey-scheme of hyper- geometric orthogonal polynomials and their q-analogues. Most formulas listed in this report can be found somewhere in the literature, but a handbook containing all these formulas did not exist. We collected known formulas for these hypergeometric orthogonal polynomials and we arranged them into the Askey-scheme and into a q-analogue of this scheme which we called the q-scheme. Thisq-schemewasnotcompletelydocumentedintheliterature. Sowefilledinsomegapsinorder to get some sort of ‘complete’ scheme of q-hypergeometric orthogonal polynomials. In chapter 0 we give some general definitions and formulas which can be used to transform several formulas into different forms of the same formula. In the other chapters we used the most common notations, but sometimes we had to change some notations in order to be consistent. For each family of orthogonal polynomials listed in this report we give conditions on the pa- rameters for which the corresponding weight function is positive. These conditions are mentioned in the orthogonality relations. We remark that many of these orthogonal polynomials are still polynomials for other values of the parameters and that they can be defined for other values as well. That is why we gave no restrictions in the definitions. As pointed out in chapter 0 some definitions can be transformed into different forms so that they are valid for some values of the parameters for which the given form has no meaning. Other formulas, such as the generating functions, are only valid for some special values of parameters and arguments. These conditions are mostly left out in this report. We are aware of the fact that this report is by no means a full description of all that is known about (basic) hypergeometric orthogonal polynomials. More on each listed family of orthogonal polynomials can be found in the articles and books to which we refer. Roelof Koekoek and Ren´e F. Swarttouw. Roelof Koekoek Ren´e F. Swarttouw Delft University of Technology Free University of Amsterdam Faculty of Technical Mathematics and Informatics Faculty of Mathematics and Informatics Mekelweg 4 De Boelelaan 1081 2628 CD Delft 1081 HV Amsterdam The Netherlands The Netherlands [email protected] [email protected] http://aw.twi.tudelft.nl/∼koekoek/ http://www.cs.vu.nl/∼rene/ 5 6 Definitions and miscellaneous formulas 0.1 Introduction In this report we will list all known sets of orthogonal polynomials which can be defined in terms of a hypergeometric function or a basic hypergeometric function. In the first part of the report we give a description of all classical hypergeometric orthogonal polynomials which appear in the so-called Askey-scheme. We give definitions, orthogonality re- lations, three term recurrence relations, second order differential or difference equations, forward andbackwardshiftoperators,Rodrigues-typeformulasandgeneratingfunctionsforallfamiliesof orthogonal polynomials listed in this Askey-scheme of hypergeometric orthogonal polynomials. In the second part we obtain a q-analogue of this scheme. We give definitions, orthogonality relations,threetermrecurrencerelations,secondorderdifferenceequations,forwardandbackward shiftoperators,Rodrigues-typeformulasandgeneratingfunctionsforallknownq-analoguesofthe hypergeometric orthogonal polynomials listed in the Askey-scheme. Further we give limit relations between different families of orthogonal polynomials in both schemesandwepointouthowtoobtaintheclassicalhypergeometricorthogonalpolynomialsfrom their q-analogues. The theory of q-analogues or q-extensions of classical formulas and functions is based on the observation that 1−qα lim =α. q→1 1−q Therefore the number (1−qα)/(1−q) is sometimes called the basic number [α]. Now we can give a q-analogue of the Pochhammer-symbol (a) which is defined by k (a) :=1 and (a) :=a(a+1)(a+2)···(a+k−1), k =1,2,3,.... 0 k This q-extension is given by (a;q) :=1 and (a;q) :=(1−a)(1−aq)(1−aq2)···(1−aqk−1), k =1,2,3,.... 0 k It is clear that (qα;q) lim k =(α) . q→1(1−q)k k In this report we will always assume that 0<q <1. For more details concerning the q-theory the reader is referred to the book [193] by G. Gasper and M. Rahman. Since many formulas given in this report can be reformulated in many different ways we will give a selection of formulas , which can be used to obtain other forms of definitions, orthogonality relations and generating functions. Most of these formulas given in this chapter can be found in [193]. We remark that in orthogonality relations we often have to add some condition(s) on the parameters of the orthogonal polynomials involved in order to have positive weight functions. By 7 using the famous theorem of Favard these conditions can also be obtained from the three term recurrence relation. In some cases, however, some conditions on the parameters may be needed in other formulas too. For instance, the definition (1.11.1) of the Laguerre polynomials has no meaning for negative integer values of the parameter α. But in fact the Laguerre polynomials are also polynomials in the parameter α. This can be seen by writing n L(α)(x)= 1 X(−n)k(α+k+1) xk. n n! k! n−k k=0 In this way the Laguerre polynomials are defined for all values of the parameter α. A similar remark holds for the Jacobi polynomials given by (1.8.1). We may also write (see section 0.4 for the definition of the hypergeometric function F ) 2 1 (cid:18) (cid:12) (cid:19) Pn(α,β)(x)=(−1)n(β+n!1)n2F1 −n,n+β+α+1 β+1(cid:12)(cid:12)(cid:12)1+2 x , which implies the well-known symmetry relation P(α,β)(x)=(−1)nP(β,α)(−x). n n Even more general we have P(α,β)(x)= 1 Xn (−n)k(n+α+β+1) (α+k+1) (cid:18)1−x(cid:19)k. n n! k! k n−k 2 k=0 FromthisformitisclearthattheJacobipolynomialscanbedefinedforallvaluesoftheparameters α and β although the definition (1.8.1) is not valid for negative integer values of the parameter α. We will not indicate these difficulties in each formula. Finally,weremarkthatineachrecurrencerelationlistedinthisreport,exceptfor(1.8.34)and (1.8.36) for the Chebyshev polynomials of the first kind, we may use P (x) = 0 and P (x) = 1 −1 0 as initial conditions. 0.2 The q-shifted factorials The symbols (a;q) defined in the preceding section are called q-shifted factorials. They can also k be defined for negative values of k as 1 (a;q) := , a6=q,q2,q3,...,q−k, k =−1,−2,−3,.... (0.2.1) k (1−aq−1)(1−aq−2)···(1−aqk) Now we have 1 (−qa−1)n (n) (a;q)−n = (aq−n;q) = (qa−1;q) q 2 , n=0,1,2,..., (0.2.2) n n where (cid:18) (cid:19) n n(n−1) = . 2 2 We can also define ∞ Y (a;q) = (1−aqk). ∞ k=0 This implies that (a;q) (a;q) = ∞ , (0.2.3) n (aqn;q) ∞ 8