Full orthogonality of bilateral q-ultraspherical functions

Establish a full orthogonality relation for the bilateral q-ultraspherical functions C_n(x; β, γ | q), which are defined in terms of bilateral basic hypergeometric 2ψ2 series and depend on parameters β and γ and base q, by identifying a positive measure on the interval [-1, 1] with respect to which ∫_{-1}^1 C_m(x; β, γ | q) C_n(x; β, γ | q) w(x) dx = 0 for m ≠ n and finite for m = n.

Background

The continuous q-ultraspherical polynomials, introduced by Rogers and extensively developed by Askey and Ismail, possess a classical orthogonality relation with respect to a positive measure and satisfy numerous structural properties (generating functions, recurrence relations, and action under the Askey–Wilson operator).

This paper introduces a bilateral extension—bilateral q-ultraspherical functions—defined via specific 2ψ2 bilateral basic hypergeometric series, and establishes several parallel properties: a bilateral generating function, a three-term recurrence, and behavior under the Askey–Wilson divided difference operator. Despite these advances, a complete orthogonality theory analogous to that of the continuous q-ultraspherical polynomials was not obtained; instead, the paper proves a weaker “shifted orthogonality” property. The unresolved issue is to determine whether a true orthogonality (with a positive measure) exists for this bilateral family.