Dear All,
The next talk will be delivered by Michael Schlosser, Professor in the 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.
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