The family of confluent Virasoro fusion kernels and a non-polynomial $q$-Askey scheme

Abstract

We study the recently introduced family of confluent Virasoro fusion kernels ${\mathcal{C}}_k(b,\mathbf{\theta },{\sigma }_s,\nu )$. We study their eigenfunction properties and show that they can be viewed as non-polynomial generalizations of both the continuous dual $q$-Hahn and the big $q$-Jacobi polynomials. More precisely, we prove that: (i) ${\mathcal{C}}_k$ is a joint eigenfunction of four different difference operators for any positive integer $k$, (ii) ${\mathcal{C}}_k$ degenerates to the continuous dual $q$-Hahn polynomials when $\nu$ is suitably discretized, and (iii) ${\mathcal{C}}_k$ degenerates to the big $q$-Jacobi polynomials when ${\sigma }_s$ is suitably discretized. These observations lead us to propose the existence of a non-polynomial generalization of the $q$-Askey scheme. The top member of this non-polynomial scheme is the Virasoro fusion kernel (or, equivalently, Ruijsenaars' hypergeometric function), and its first confluence is given by the ${\mathcal{C}}_k$.

30 pages