Chapter 18 Orthogonal Polynomials

T. H. Koornwinder University of Amsterdam, Korteweg–de Vries Institute, Amsterdam, The Netherlands. R. Wong City University of Hong Kong, Liu Bie Ju Centre for Mathematical Sciences, Kowloon, Hong Kong. R. Koekoek Delft University of Technology, Delft Institute of Applied Mathematics, Delft, The Netherlands. R. F. Swarttouw Delft University of Technology, Delft Institute of Applied Mathematics, Delft, The Netherlands. W. P. Reinhardt University of Washington, Department of Chemistry, Seattle, Washington, USA.

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Notes:

The main references for writing this chapter are Andrews et al. (1999), Askey and Wilson (1985), Chihara (1978), Koekoek et al. (2010), and Szegő (1975).

Referenced by:

§14.7(i), §16.7, §3.11(ii), DLMF Update; Version 1.2.0, Profile René F. Swarttouw, Profile Roelof Koekoek, Profile Roderick S. C. Wong, Profile Roderick S. C. Wong, Profile Tom H. Koornwinder, Profile Wolter Groenevelt, § ‣ Chapter Authors, § ‣ Chapter Authors, § ‣ Chapter Authors, § ‣ Chapter Authors, § ‣ Associate Editors, § ‣ Associate Editors, § ‣ Contributing Developers, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials, Erratum (V1.1.6) for Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols, Erratum (V1.1.6) for Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions, Erratum (V1.2.0) for Chapter 18 Additions, Version 1.2.0 (March 27, 2024), Version 1.2.0 (March 27, 2024), Organization and Objective, § ‣ Software Cross Index

Permalink:

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• Notation
  1. 18.1 Notation
• General Orthogonal Polynomials
  1. 18.2 General Orthogonal Polynomials
• Classical Orthogonal Polynomials
  1. 18.3 Definitions
  2. 18.4 Graphics
  3. 18.5 Explicit Representations
  4. 18.6 Symmetry, Special Values, and Limits to Monomials
  5. 18.7 Interrelations and Limit Relations
  6. 18.8 Differential Equations
  7. 18.9 Recurrence Relations and Derivatives
  8. 18.10 Integral Representations
  9. 18.11 Relations to Other Functions
  10. 18.12 Generating Functions
  11. 18.13 Continued Fractions
  12. 18.14 Inequalities
  13. 18.15 Asymptotic Approximations
  14. 18.16 Zeros
  15. 18.17 Integrals
  16. 18.18 Sums
• Askey Scheme
  1. 18.19 Hahn Class: Definitions
  2. 18.20 Hahn Class: Explicit Representations
  3. 18.21 Hahn Class: Interrelations
  4. 18.22 Hahn Class: Recurrence Relations and Differences
  5. 18.23 Hahn Class: Generating Functions
  6. 18.24 Hahn Class: Asymptotic Approximations
  7. 18.25 Wilson Class: Definitions
  8. 18.26 Wilson Class: Continued

• Other Orthogonal Polynomials
  1. 18.27 $q$-Hahn Class
  2. 18.28 Askey–Wilson Class
  3. 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
  4. 18.30 Associated OP’s
  5. 18.31 Bernstein–Szegő Polynomials
  6. 18.32 OP’s with Respect to Freud Weights
  7. 18.33 Polynomials Orthogonal on the Unit Circle
  8. 18.34 Bessel Polynomials
  9. 18.35 Pollaczek Polynomials
  10. 18.36 Miscellaneous Polynomials
  11. 18.37 Classical OP’s in Two or More Variables
• Applications
  1. 18.38 Mathematical Applications
  2. 18.39 Applications in the Physical Sciences
• Computation
  1. 18.40 Methods of Computation
  2. 18.41 Tables
  3. 18.42 Software

17.19 Software18.1 Notation

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