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

TL;DR

This paper introduces a family of confluent Virasoro fusion kernels that generalize classical $q$-orthogonal polynomials and proposes a new non-polynomial $q$-Askey scheme, linking special functions with conformal field theory.

Contribution

It demonstrates that confluent Virasoro fusion kernels are eigenfunctions of difference operators and generalize known $q$-polynomials, suggesting a novel non-polynomial $q$-Askey scheme.

Findings

Fusion kernels are eigenfunctions of four difference operators.

They degenerate to continuous dual $q$-Hahn polynomials under discretization.

They degenerate to big $q$-Jacobi polynomials under different discretization.

Abstract

We study the recently introduced family of confluent Virasoro fusion kernels $\mathcal{C}_k(b,\boldsymbol{\theta},\sigma_s,\nu)$. 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 $\mathcal{C}_k$.