## Hypergeometric Orthogonal Polynomials and Their q-AnaloguesThe present book is about the Askey scheme and the q-Askey scheme, which are graphically displayed right before chapter 9 and chapter 14, respectively. The fa- lies of orthogonal polynomials in these two schemes generalize the classical orth- onal polynomials (Jacobi, Laguerre and Hermite polynomials) and they have pr- erties similar to them. In fact, they have properties so similar that I am inclined (f- lowing Andrews & Askey [34]) to call all families in the (q-)Askey scheme classical orthogonal polynomials, and to call the Jacobi, Laguerre and Hermite polynomials very classical orthogonal polynomials. These very classical orthogonal polynomials are good friends of mine since - most the beginning of my mathematical career. When I was a fresh PhD student at the Mathematical Centre (now CWI) in Amsterdam, Dick Askey spent a sabbatical there during the academic year 1969–1970. He lectured to us in a very stimulating wayabouthypergeometricfunctionsandclassicalorthogonalpolynomials. Evenb- ter, he gave us problems to solve which might be worth a PhD. He also pointed out to us that there was more than just Jacobi, Laguerre and Hermite polynomials, for instance Hahn polynomials, and that it was one of the merits of the Higher Transc- dental Functions (Bateman project) that it included some newer stuff like the Hahn polynomials (see [198, §10. 23]). |

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### Contents

1 | |

Polynomial Solutions of Eigenvalue Problems | 28 |

Orthogonality of the Polynomial Solutions | 53 |

Part I Classical Orthogonal Polynomials | 76 |

Part II Classical qOrthogonal Polynomials | 254 |

553 | |

575 | |

### Other editions - View all

Hypergeometric Orthogonal Polynomials and Their q-Analogues Roelof Koekoek,Peter A. Lesky,René F. Swarttouw No preview available - 2012 |

Hypergeometric Orthogonal Polynomials and Their q-Analogues Roelof Koekoek,Peter A. Lesky,René F. Swarttouw No preview available - 2010 |

Hypergeometric Orthogonal Polynomials and Their q-Analogues Roelof Koekoek,Peter A. Lesky,René F. Swarttouw No preview available - 2010 |

### Common terms and phrases

Al-Salam-Carlitz Askey-Wilson polynomials Backward Shift Operator Basic Hypergeometric Representation Bessel polynomials big q-Jacobi boundary conditions Charlier polynomials coefficients continuous dual Hahn continuous Hahn definition difference equation differential equation discrete q-Hermite dual Hahn polynomials dual q-Hahn eigenvalue equivalently finite system Forward Shift Operator Hahn polynomials given Hence Hermite polynomials Hypergeometric Orthogonal Polynomials implies infinite Jacobi polynomials Laguerre leads Limit Relations Mathematics Meixner polynomials Meixner-Pollaczek Meixner-Pollaczek polynomials mials monic polynomials nomials Normalized Recurrence Relation obtain Ömn operator equation Orthogonal Polynomial Solutions orthogonality relation pn(x poly polyno polynomials and take Positive-Definite Orthogonal Polynomial possible solution q-analogue q-Bessel q-Charlier q-derivative q-Difference q-Meixner q-Racah q-ultraspherical Racah polynomials Recurrence Relation xpn(x regularity condition 2.3.3 Rodrigues formula Rodrigues-Type Formula system of polynomials take the limit theorem three-term recurrence relation two-term recurrence relation weight function Wilson polynomials yn(x yöq