Orthogonal Basic Hypergeometric Laurent Polynomials

Mourad E. H. Ismail ; Dennis Stanton

Symmetry, integrability and geometry: methods and applications, Tome 8 (2012)

Résumé

The Askey–Wilson polynomials are orthogonal polynomials in $x = \cos \theta$, which are given as a terminating $_4\phi_3$ basic hypergeometric series. The non-symmetric Askey–Wilson polynomials are Laurent polynomials in $z=e^{i\theta}$, which are given as a sum of two terminating $_4\phi_3$'s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single $_4\phi_3$'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.

Détail

Keywords: Askey–Wilson polynomials; orthogonality.

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