Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Preliminaries
- 2 General Orthogonal Polynomials
- 3 Jacobi and Related Polynomials
- 4 Recursively Defined Polynomials
- 5 Wilson and Related Polynomials
- 6 Discrete Orthogonal Polynomials
- 7 Some q-Orthogonal Polynomials
- 8 The Askey–Wilson Family of Polynomials
- 9 Orthogonal Polynomials on the Unit Circle
- 10 Zeros of Orthogonal Polynomials
- 11 The Moment Problem
- 12 Matrix-Valued Orthogonal Polynomials and Differential Equations
- 13 Some Families of Matrix-Valued Jacobi Orthogonal Polynomials
- References
- Index
8 - The Askey–Wilson Family of Polynomials
Published online by Cambridge University Press: 14 September 2020
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Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Preliminaries
- 2 General Orthogonal Polynomials
- 3 Jacobi and Related Polynomials
- 4 Recursively Defined Polynomials
- 5 Wilson and Related Polynomials
- 6 Discrete Orthogonal Polynomials
- 7 Some q-Orthogonal Polynomials
- 8 The Askey–Wilson Family of Polynomials
- 9 Orthogonal Polynomials on the Unit Circle
- 10 Zeros of Orthogonal Polynomials
- 11 The Moment Problem
- 12 Matrix-Valued Orthogonal Polynomials and Differential Equations
- 13 Some Families of Matrix-Valued Jacobi Orthogonal Polynomials
- References
- Index
Summary
The Al-Salam–Chihara polynomials appeared in a characterization problem regarding convolutions of orthogonal polynomials. Al-Salam and Chihara (1976) only recorded the three-term recurrence relation and a generating function. The weight function was first found by Askey and Ismail (1983, 1984), who also named the polynomials after the ones who first identified them.
A basic hypergeometric representation is
pn(x;t1,t2|q)=ϕ32(q−n,t1eiθ,t1e−iθt1t2,0|q,q), (8.1.1)
and the orthogonality relation is
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- Encyclopedia of Special Functions: The Askey-Bateman Project , pp. 178 - 198Publisher: Cambridge University PressPrint publication year: 2020