Hypergeometric Orthogonal Polynomials and Their qAnalogues
Authors: Koekoek, Roelof, Lesky, Peter A., Swarttouw, René F.
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The very classical orthogonal polynomials named after Hermite, Laguerre and Jacobi, satisfy many common properties. For instance, they satisfy a secondorder differential equation with polynomial coefficients and they can be expressed in terms of a hypergeometric function.
Replacing the differential equation by a secondorder difference equation results in (discrete) orthogonal polynomial solutions with similar properties. Generalizations of these difference equations, in terms of Hahn's qdifference operator, lead to both continuous and discrete orthogonal polynomials with similar properties. For instance, they can be expressed in terms of (basic) hypergeometric functions.
Based on Favard's theorem, the authors first classify all families of orthogonal polynomials satisfying a secondorder differential or difference equation with polynomial coefficients. Together with the concept of duality this leads to the families of hypergeometric orthogonal polynomials belonging to the Askey scheme. For each family they list the most important properties and they indicate the (limit) relations.
Furthermore the authors classify all qorthogonal polynomials satisfying a secondorder qdifference equation based on Hahn's qoperator. Together with the concept of duality this leads to the families of basic hypergeometric orthogonal polynomials which can be arranged in a qanalogue of the Askey scheme. Again, for each family they list the most important properties, the (limit) relations between the various families and the limit relations (for q > 1) to the classical hypergeometric orthogonal polynomials belonging to the Askey scheme.
These (basic) hypergeometric orthogonal polynomials have several applications in various areas of mathematics and (quantum) physics such as approximation theory, asymptotics, birth and death processes, probability and statistics, coding theory and combinatorics.
 Reviews

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)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Hypergeometric Orthogonal Polynomials and Their qAnalogues
 Authors

 Roelof Koekoek
 Peter A. Lesky
 René F. Swarttouw
 Series Title
 Springer Monographs in Mathematics
 Copyright
 2010
 Publisher
 SpringerVerlag Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 eBook ISBN
 9783642050145
 DOI
 10.1007/9783642050145
 Hardcover ISBN
 9783642050138
 Softcover ISBN
 9783642263514
 Series ISSN
 14397382
 Edition Number
 1
 Number of Pages
 XIX, 578
 Number of Illustrations and Tables
 2 b/w illustrations
 Topics