Orthogonal Basic Hypergeometric Laurent Polynomials

Abstract

The Askey-Wilson polynomials are orthogonal polynomials in $x=\cos \theta$, which are given as a terminating ${}_4{\varphi }_3$ 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 ${}_4{\varphi }_3$'s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single ${}_4{\varphi }_3$'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.

References in corpus (1)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue