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The Askey-Scheme of Hypergeometric Orthogonal Polynomials And The Askeyscheme of hyp ergeometric orthogonal p olynomials and its q analogue Ro elof Ko eko ek Rene F Swarttouw February Abstract We list the socalled Askeyscheme of hyp ergeometric orthogonal p olynomials In chapter we give the denition the orthogonality relation the three term recurrence relation and generating functions of all classes of orthogonal p olynomials in this scheme In chapter we give all limit relations b etween dierent classes of orthogonal p olynomials listed in the Askeyscheme In chapter we list the q analogues of the p olynomials in the Askeyscheme We give their denition orthogonality relation three term recurrence relation and generating functions In chapter we give the limit relations b etween those basic hyp ergeometric orthogonal p olynomials Finally in chapter we p oint out how the classical hyp ergeometric orthogonal p olynomials of the Askeyscheme can b e obtained from their q analogues Acknowledgement We would like to thank Professor Tom H Ko ornwinder who suggested us to write a rep ort like this He also help ed us solving many problems we encountered during the research and provided us with several references Contents Preface Denitions and miscellaneous formulas Intro duction The q shifted factorials The q gamma function and the q binomial co ecient Hyp ergeometric and basic hyp ergeometric functions The q binomial theorem and other summation formulas Transformation formulas Some sp ecial functions and their q analogues The q derivative and the q integral ASKEYSCHEME Hyp ergeometric orthogonal p olynomials Wilson Racah Continuous dual Hahn Continuous Hahn Hahn Dual Hahn MeixnerPollaczek Jacobi Gegenbauer Ultraspherical Chebyshev Legendre Spherical Meixner Krawtchouk Laguerre Charlier Hermite Limit relations b etween hyp ergeometric orthogonal p olynomials Wilson Continuous dual Hahn Wilson Continuous Hahn Wilson Jacobi Racah Hahn Racah Dual Hahn Continuous dual Hahn MeixnerPollaczek Continuous Hahn MeixnerPollaczek Continuous Hahn Jacobi Hahn Jacobi Hahn Meixner Hahn Krawtchouk Dual Hahn Meixner Dual Hahn Krawtchouk MeixnerPollaczek Laguerre MeixnerPollaczek Hermite Jacobi Laguerre Jacobi Hermite Meixner Laguerre Meixner Charlier Krawtchouk Charlier Krawtchouk Hermite Laguerre Hermite Charlier Hermite q SCHEME Basic hyp ergeometric orthogonal p olynomials AskeyWilson q Racah Continuous dual q Hahn Continuous q Hahn Big q Jacobi Big q Legendre q Hahn Dual q Hahn AlSalamChihara q MeixnerPollaczek Continuous q Jacobi Continuous q ultraspherical Rogers Continuous q Legendre Big q Laguerre Little q Jacobi Little q Legendre q Meixner Quantum q Krawtchouk q Krawtchouk Ane q Krawtchouk Dual q Krawtchouk Continuous big q Hermite Continuous q Laguerre Little q Laguerre Wall q Laguerre Alternative q Charlier q Charlier AlSalamCarlitz I AlSalamCarlitz I I Continuous q Hermite StieltjesWigert Discrete q Hermite I Discrete q Hermite I I Limit relations b etween basic hyp ergeometric orthogonal p olynomials AskeyWilson Continuous dual q Hahn AskeyWilson Continuous q Hahn AskeyWilson Big q Jacobi AskeyWilson Continuous q Jacobi AskeyWilson Continuous q ultraspherical Rogers q Racah Big q Jacobi .
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