Orthogonal basic hypergeometric Laurent polynomials

Mourad E.H. Ismail; Dennis Stanton

doi:10.3842/SIGMA.2012.092

Orthogonal basic hypergeometric Laurent polynomials

Mourad E.H. Ismail, Dennis Stanton

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

The Askey-Wilson polynomials are orthogonal polynomials in x = cos θ, which are given as a terminating 4φ3 basic hypergeometric series. The non-symmetric Askey- Wilson polynomials are Laurent polynomials in z = e^iθ, which are given as a sum of two terminating 4φ3's. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single 4φ3's which are Laurent polynomials in z are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetric Askey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques.

Original language	English (US)
Article number	092
Journal	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume	8
DOIs	https://doi.org/10.3842/SIGMA.2012.092
State	Published - 2012

Keywords

Askey-wilson polynomials
Orthogonality

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abstract = "The Askey-Wilson polynomials are orthogonal polynomials in x = cos θ, which are given as a terminating 4φ3 basic hypergeometric series. The non-symmetric Askey- Wilson polynomials are Laurent polynomials in z = eiθ, which are given as a sum of two terminating 4φ3's. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single 4φ3's which are Laurent polynomials in z are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetric Askey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques.",

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AU - Ismail, Mourad E.H.

AU - Stanton, Dennis

PY - 2012

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N2 - The Askey-Wilson polynomials are orthogonal polynomials in x = cos θ, which are given as a terminating 4φ3 basic hypergeometric series. The non-symmetric Askey- Wilson polynomials are Laurent polynomials in z = eiθ, which are given as a sum of two terminating 4φ3's. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single 4φ3's which are Laurent polynomials in z are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetric Askey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques.

AB - The Askey-Wilson polynomials are orthogonal polynomials in x = cos θ, which are given as a terminating 4φ3 basic hypergeometric series. The non-symmetric Askey- Wilson polynomials are Laurent polynomials in z = eiθ, which are given as a sum of two terminating 4φ3's. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single 4φ3's which are Laurent polynomials in z are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetric Askey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques.

KW - Askey-wilson polynomials

KW - Orthogonality

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