Dear All,
The next talk will be delivered by Michael Schlosser, Department of Mathematics at the University of Vienna. Please note the changed time for the coming semester.
Talk Announcement:
Title: Bilateral $q$-ultraspherical functions
Title: Bilateral $q$-ultraspherical functions
Speaker: Michael Schlosser (University of Vienna).
When: August 20, 2026, 5:00 PM- 6:00 PM IST
When: August 20, 2026, 5:00 PM- 6:00 PM IST
Where: Zoom: Write to the organisers for the link
Abstract: We introduce the bilateral $q$-ultraspherical functions $C_n(x;\beta,\gamma|q)$ (where $n$ can be any integer), a bilateral-series extension of the continuous $q$-ultraspherical polynomials $C_n(x;\beta|q)$ (where $n$ is a nonnegative integer). 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 $q$. Our main results for the bilateral $q$-ultraspherical functions $C_n(x;\beta,\gamma|q)$ include full orthogonality relations with respect to explicit orthogonality functionals involving analytic mass aggregates, which we establish by analytic continuation. Other results satisfied by the $C_n(x;\beta,\gamma|q)$ include a product formula for their bilateral generating function, a three-term recurrence relation, their transformation under the Askey--Wilson divided difference operator, Rodrigues-type formulae, and explicit large-order asymptotic expansions. We also obtain shifted orthogonality relations and a bilateral Chen--Liu type mixed orthogonality formula. In the limit $\gamma\to 1$, our results for the bilateral $q$-ultraspherical functions $C_n(x;\beta,\gamma|q)$ reduce to classical results for the continuous $q$-ultraspherical polynomials $C_n(x;\beta|q)$. Full details are given in https://arxiv.org/abs/2508.
