Hypergeometric Orthogonal Polynomials and Their q-Analogues
The very classical orthogonal polynomials named after Hermite, Laguerre and Jacobi, satisfy many common properties. For instance, they satisfy a second-order differential equation with polynomial coefficients and they can be expressed in terms of a hyperg
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Roelof Koekoek r Peter A. Lesky Ren´e F. Swarttouw
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Hypergeometric Orthogonal Polynomials and Their q-Analogues
With a Foreword by Tom H. Koornwinder
Roelof Koekoek Delft Institute of Applied Mathematics Delft University of Technology P.O. Box 5031 2600 GA Delft The Netherlands [email protected] Peter A. Lesky (1927–2008) University of Stuttgart, Germany
Ren´e F. Swarttouw Department of Mathematics Faculty of Sciences VU University Amsterdam De Boelelaan 1081a 1081 HV Amsterdam The Netherlands [email protected] Foreword by Tom H. Koornwinder Korteweg-de Vries Institute of Mathematics Faculty of Science University of Amsterdam P.O. Box 94248 1090 GE Amsterdam The Netherlands [email protected]
ISSN 1439-7382 ISBN 978-3-642-05013-8 e-ISBN 978-3-642-05014-5 DOI 10.1007/978-3-642-05014-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010923797 Mathematics Subject Classification (2000): 33C45; 33D45 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover illustration: The Authors Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Foreword
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 families of orthogonal polynomials in these two schemes generalize the classical orthogonal polynomials (Jacobi, Laguerre and Hermite polynomials) and they have properties similar to them. In fact, they have properties so similar that I am inclined (following 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 almost 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 way about hypergeometric functions and classical orthogonal polynomials. Ev
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