Bilateral $q$-ultraspherical functions

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.