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Hypergeometric Orthogonal Polynomials and Their q-Analogues

  • Book
  • © 2010

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Authors:
  1. Roelof Koekoek

    1. Fac. Mathematics & Informatics, Delft University of Technology, Delft, Netherlands

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  2. Peter A. Lesky

    1. Fakultät für Mathematik, Universität Stuttgart, Stuttgart, Germany

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  3. René F. Swarttouw

    1. Dept. Mathematics &, Free University Amsterdam, Amsterdam, Netherlands

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Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (14 chapters)

  1. Front Matter

    Pages I-XIX
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  2. Definitions and Miscellaneous Formulas

    • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
    Pages 1-27

  3. Polynomial Solutions of Eigenvalue Problems

    • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
    Pages 29-51

  4. Orthogonality of the Polynomial Solutions

    • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
    Pages 53-75

  5. Classical Orthogonal Polynomials

    1. Front Matter

      Pages 77-77
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  6. Classical orthogonal polynomials

    1. Orthogonal Polynomial Solutions of Differential Equations

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 79-93

    2. Orthogonal Polynomial Solutions of Real Difference Equations

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 95-121

    3. Orthogonal Polynomial Solutions of Complex Difference Equations

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 123-139

    4. Orthogonal Polynomial Solutions in x(x+u) of Real Difference Equations

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 141-170

    5. Orthogonal Polynomial Solutions in z(z+u) of Complex Difference Equations

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 171-181

    6. Hypergeometric Orthogonal Polynomials

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 183-253

  7. Classical q-Orthogonal Polynomials

    1. Front Matter

      Pages 255-255
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  8. Classical q-orthogonal polynomials

    1. Orthogonal Polynomial Solutions of q-Difference Equations

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 257-322

    2. Orthogonal Polynomial Solutions in q −x of q-Difference Equations

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 323-367

    3. Orthogonal Polynomial Solutions in q −x+uq x of Real q-Difference Equations

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 369-394

    4. Orthogonal Polynomial Solutions in \(\frac{a}{z}+\frac{uz}{a}\) of Complex q-Difference Equations

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 395-411

    5. Basic Hypergeometric Orthogonal Polynomials

      • Roelof Koekoek, Peter A. Lesky, René F. Swarttouw
      Pages 413-552

  9. Back Matter

    Pages 553-578
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Keywords

About this book

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]).

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)

Authors and Affiliations

  • Fac. Mathematics & Informatics, Delft University of Technology, Delft, Netherlands

    Roelof Koekoek

  • Fakultät für Mathematik, Universität Stuttgart, Stuttgart, Germany

    Peter A. Lesky

  • Dept. Mathematics &, Free University Amsterdam, Amsterdam, Netherlands

    René F. Swarttouw

Bibliographic Information

  • Book Title: Hypergeometric Orthogonal Polynomials and Their q-Analogues

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

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-05014-5

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2010

  • Hardcover ISBN: 978-3-642-05013-8Published: 30 May 2010

  • Softcover ISBN: 978-3-642-26351-4Published: 28 June 2012

  • eBook ISBN: 978-3-642-05014-5Published: 18 March 2010

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XIX, 578

  • Number of Illustrations: 2 b/w illustrations

  • Topics: Analysis, Special Functions

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