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Springer Monographs in Mathematics

Hypergeometric Orthogonal Polynomials and Their q-Analogues

Authors: Koekoek, Roelof, Lesky, Peter A., Swarttouw, René F.

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  • ISBN 978-3-642-05014-5
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Softcover $129.00
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  • ISBN 978-3-642-26351-4
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About this book

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 hypergeometric function.

Replacing the differential equation by a second-order difference equation results in (discrete) orthogonal polynomial solutions with similar properties. Generalizations of these difference equations, in terms of Hahn's q-difference 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 second-order 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 q-orthogonal polynomials satisfying a second-order q-difference equation based on Hahn's q-operator. Together with the concept of duality this leads to the families of basic hypergeometric orthogonal polynomials which can be arranged in a q-analogue 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)

Table of contents (14 chapters)

  • Definitions and Miscellaneous Formulas

    Koekoek, Roelof (et al.)

    Pages 1-27

  • Polynomial Solutions of Eigenvalue Problems

    Koekoek, Roelof (et al.)

    Pages 29-51

  • Orthogonality of the Polynomial Solutions

    Koekoek, Roelof (et al.)

    Pages 53-75

  • Orthogonal Polynomial Solutions of Differential Equations

    Koekoek, Roelof (et al.)

    Pages 79-93

  • Orthogonal Polynomial Solutions of Real Difference Equations

    Koekoek, Roelof (et al.)

    Pages 95-121

Buy this book

eBook $99.00
price for USA
  • ISBN 978-3-642-05014-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $129.00
price for USA
  • ISBN 978-3-642-05013-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $129.00
price for USA
  • ISBN 978-3-642-26351-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Hypergeometric Orthogonal Polynomials and Their q-Analogues
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2010
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-05014-5
DOI
10.1007/978-3-642-05014-5
Hardcover ISBN
978-3-642-05013-8
Softcover ISBN
978-3-642-26351-4
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XIX, 578
Number of Illustrations and Tables
2 b/w illustrations
Topics