Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 6

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

Pages: 1597 – 1650

DOI: https://dx.doi.org/10.4310/ATMP.2021.v25.n6.a5

Authors

Jonatan Lenells (Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden)

Julien Roussillon (Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden)

Abstract

We study the recently introduced family of confluent Virasoro fusion kernels $\mathcal{C}_k (b, \theta, \sigma_s, \nu)$. 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 nonpolynomial scheme is the Virasoro fusion kernel (or, equivalently, Ruijsenaars’ hypergeometric function), and its first confluence is given by the $\mathcal{C}_k$.

Published 24 June 2022