Superintegrability of the Wilson family of matrix models and moments of multivariable orthogonal polynomials

TL;DR

This paper introduces new superintegrable matrix models linked to multivariate orthogonal polynomials, revealing novel phenomena and providing explicit moment expressions for these models.

Contribution

It establishes new superintegrable matrix models based on multivariate Wilson and Meixner-Pollaczek polynomials, expanding the understanding of their algebraic and combinatorial structures.

Findings

New superintegrable matrix models based on multivariate polynomials

Explicit formulas for moments of multivariate measures

Discovery of phenomena deviating from classical Schur polynomial basis

Abstract

We present new examples of superintegrable matrix/eigenvalue models. These examples arise as a result of the exploration of the relationship between the theory of superintegrability and multivariate orthogonal polynomials. The new superintegrable examples are built upon the multivariate generalizations of the Meixner-Pollaczek and Wilson polynomials and their respective measures. From the perspective of multivariate orthogonal polynomials in this work we propose expressions for (generalized) moments of the respective multi-variable measures. From the perspective of superintegrability we uncover a couple of new phenomena such as the deviation from Schur polynomials as the superintegrable basis without any deformation and new combinatorial structures appearing in the answers.