{"title":"The family of confluent Virasoro fusion kernels and a non-polynomial $q$-Askey scheme","authors":"J. Lenells, J. Roussillon","doi":"10.4310/atmp.2021.v25.n6.a5","DOIUrl":null,"url":null,"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 $\\mathcal{C}_k$.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Theoretical and Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.4310/atmp.2021.v25.n6.a5","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 1
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 $\mathcal{C}_k$.
期刊介绍:
Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.