Orthogonal Basic Hypergeometric Laurent Polynomials

TL;DR

This paper introduces new orthogonality relations for Laurent polynomial forms of basic hypergeometric series, extending the understanding of non-symmetric Askey-Wilson polynomials through classical analysis methods.

Contribution

It provides novel orthogonality relations for $_4\phi_3$ Laurent polynomials, including discrete cases, connecting classical analysis with previous algebraic results.

Findings

New orthogonality relations for $_4\phi_3$ Laurent polynomials

Biorthogonality of non-symmetric Askey-Wilson polynomials

Inclusion of discrete orthogonality relations

Abstract

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.