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
7 - Some q-Orthogonal 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 continuous q-ultraspherical and continuous q-Hermite polynomials first appeared in Rogers’ work on the Rogers–Ramanujan identities in 1893–95 (Askey and Ismail, 1983). They belong to the Fejér class of polynomials having a generating function of the form
∑n=0∞ϕn(cosθ)tn=|F(reiθ)|2, (7.0.1)
where F(z) is analytic in a neighborhood of z=0. Feldheim (1941) and Lanzewizky (1941) independently proved that the only orthogonal generalized polynomials in the Fejér class are either the ultraspherical polynomials or the q-ultraspherical polynomials or special cases of them. They proved that F has to be F1 or F2, or some limiting cases of them, where
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- Encyclopedia of Special Functions: The Askey-Bateman Project , pp. 157 - 177Publisher: Cambridge University PressPrint publication year: 2020