BASIC HYPERGEOMETRIC FUNCTIONS and ORTHOGONAL LAURENT POLYNOMIALS

Abstract

A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or q-hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials {(2)Phi(1)(q(-n), q(b+1); q(-c+b-n); q,q(-c+d-1)z)}(n=0)(infinity), where 0 < q < 1 and the complex parameters b, c and d are such that b not equal -1, -2, ... , c - b + 1 not equal -1, -2, ... , Re(d) > 0 and Re(c - d + 2) > 0. Explicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szego polynomials are also derived.