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arXiv · 2508.08908

Bilateral $q$-ultraspherical functions

Michael J. Schlosser
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Abstract

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 $\gamma\to1$, the construction and identities reduce to the classical results for the continuous $q$-ultraspherical polynomials.

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Michael J. Schlosser. 2025-08-12. Bilateral $q$-ultraspherical functions. https://arxiv.org/abs/2508.08908

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