Hypergeometric Orthogonal Polynomials and Their q-Analogues

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

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3. Polynomial Solutions of Eigenvalue Problems

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

   Pages 29-51

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4. Orthogonality of the Polynomial Solutions

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

   Pages 53-75

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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

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   2. Orthogonal Polynomial Solutions of Real Difference Equations

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

      Pages 95-121

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   3. Orthogonal Polynomial Solutions of Complex Difference Equations

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

      Pages 123-139

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

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

      Pages 141-170

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

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

      Pages 171-181

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   6. Hypergeometric Orthogonal Polynomials

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

      Pages 183-253

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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

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

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

      Pages 323-367

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   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

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   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

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   5. Basic Hypergeometric Orthogonal Polynomials

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

      Pages 413-552

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9. Back Matter

   Pages 553-578

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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]).

• Askey scheme
• Eigenvalue
• Hypergeometric function
• basic hypergeometric functions
• differential equation
• hypergeometric functions
• orthogonal polynomials
• q-orthogonal polynomials

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)

• 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

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