Hypergeometric Orthogonal Polynomials and Their q-Analogues
Seiten
2012
|
2010
Springer Berlin (Verlag)
978-3-642-26351-4 (ISBN)
Springer Berlin (Verlag)
978-3-642-26351-4 (ISBN)
This book classifies all families of orthogonal polynomials satisfying a second-order differential or difference equation with polynomial coefficients, which leads to the families of hypergeometric orthogonal polynomials belonging to the Askey scheme.
The 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]).
The 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]).
Definitions and Miscellaneous Formulas.- Classical orthogonal polynomials.- Orthogonal Polynomial Solutions of Differential Equations.- Orthogonal Polynomial Solutions of Real Difference Equations.- Orthogonal Polynomial Solutions of Complex Difference Equations.- Orthogonal Polynomial Solutions in x(x+u) of Real Difference Equations.- Orthogonal Polynomial Solutions in z(z+u) of Complex Difference Equations.- Hypergeometric Orthogonal Polynomials.- Polynomial Solutions of Eigenvalue Problems.- Classical q-orthogonal polynomials.- Orthogonal Polynomial Solutions of q-Difference Equations.- Orthogonal Polynomial Solutions in q?x of q-Difference Equations.- Orthogonal Polynomial Solutions in q?x+uqx of Real
From the reviews:
"The book starts with a brief but valuable foreword by Tom Koornwinder on the history of the classification problem for orthogonal polynomials. ... the ideal text for a graduate course devoted to the classification, and it is a valuable reference, which everyone who works in orthogonal polynomials will want to own." (Warren Johnson, The Mathematical Association of America, August, 2010)| Erscheint lt. Verlag | 28.6.2012 |
|---|---|
| Reihe/Serie | Springer Monographs in Mathematics |
| Vorwort | Tom H. Koornwinder |
| Zusatzinfo | XIX, 578 p. 2 illus. |
| Verlagsort | Berlin |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 876 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Schlagworte | 33C45; 33D45 • Askey scheme • basic hypergeometric functions • differential equation • eigenvalue • hypergeometric function • hypergeometric functions • Orthogonale Gruppen • orthogonal polynomials • Polynom • q-orthogonal polynomials |
| ISBN-10 | 3-642-26351-8 / 3642263518 |
| ISBN-13 | 978-3-642-26351-4 / 9783642263514 |
| Zustand | Neuware |
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