Bilateral $q$-ultraspherical functions

TL;DR

This paper introduces bilateral $q$-ultraspherical functions, extending classical polynomials with new hypergeometric series, and explores their properties including generating functions, recurrence relations, and orthogonality.

Contribution

It presents the first bilateral extension of continuous $q$-ultraspherical polynomials, detailing their fundamental properties and behaviors.

Findings

Derived a bilateral generating function for the functions

Established a three-term recurrence relation

Showed they satisfy shifted orthogonality

Abstract

We introduce a bilateral extension of the continuous $q$-ultraspherical polynomials which we call bilateral $q$-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ${}_2\psi_2$ series, they are analytic in a variable $x=\cos\theta$ and depend on two parameters $\beta$ and $\gamma$ and on a base $q$. For these bilateral $q$-ultraspherical functions we derive a bilateral generating function, find a three-term recurrence relation, explain how they behave under the action of the Askey--Wilson divided difference operator, and show that they satisfy a type of shifted orthogonality.