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

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Advances in Theoretical and Mathematical Physics Pub Date : 2020-11-16 DOI:10.4310/atmp.2021.v25.n6.a5
J. Lenells, J. Roussillon
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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 $\mathcal{C}_k$.
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收敛Virasoro融合核族和非多项式$q$-Askey格式
我们研究了最近引入的汇合Virasoro融合核族$\mathcal{C}_k(b,\boldsymbol{\theta},\sigma_s,\nu)$。我们研究了它们的特征函数性质,并证明它们可以看作是连续对偶$q$ -Hahn多项式和大的$q$ -Jacobi多项式的非多项式推广。更准确地说,我们证明了:(i) $\mathcal{C}_k$是任意正整数$k$的四种不同差分算子的联合特征函数,(ii) $\mathcal{C}_k$在$\nu$适当离散时退化为连续对偶$q$ -Hahn多项式,(iii) $\mathcal{C}_k$在$\sigma_s$适当离散时退化为大$q$ -Jacobi多项式。这些观察结果使我们提出$q$ -Askey方案的非多项式推广的存在性。该非多项式格式的顶层成员是Virasoro融合核(或等价的rujsenaars的超几何函数),其第一次汇合由$\mathcal{C}_k$给出。
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来源期刊
Advances in Theoretical and Mathematical Physics
Advances in Theoretical and Mathematical Physics 物理-物理:粒子与场物理
CiteScore
2.20
自引率
6.70%
发文量
0
审稿时长
>12 weeks
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.
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