Bilateral $q$-ultraspherical functions

We introduce the bilateral $q$-ultraspherical functions, a bilateral-series extension of the continuous $q$-ultraspherical polynomials. They are defined by specific bilateral basic hypergeometric ${}_2{\psi }_2$ series, are analytic in the variable $x=\cos \theta$, and depend on two parameters $\beta$ and $\gamma$ and on a base $q$. We derive a product formula for their bilateral generating function, a three-term recurrence relation, their transformation under the Askey–Wilson divided difference operator, three weight-based Rodrigues-type formulae, and explicit large-order asymptotic expansions. The main results are full orthogonality relations with respect to explicit orthogonality functionals involving analytic mass aggregates. We also obtain shifted orthogonality relations and a bilateral Chen–Liu type mixed orthogonality formula. In the limit $\gamma \to 1$, the construction and identities reduce to the classical results for the continuous $q$-ultraspherical polynomials.