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

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.