On diagonalizable quantum weighted Hankel matrices

Abstract

A semi-infinite weighted Hankel matrix with entries defined in terms of basic hypergeometric series is explicitly diagonalized as an operator on ${\ell }^2({\mathbb{N}}_0)$. The approach uses the fact that the operator commutes with a diagonalizable Jacobi operator corresponding to Al-Salam-Chihara orthogonal polynomials. Yet another weighted Hankel matrix, which commutes with a Jacobi operator associated with the continuous $q$-Laguerre polynomials, is diagonalized. As an application, several new integral formulas for selected quantum orthogonal polynomials are deduced. In addition, an open research problem concerning a quantum Hilbert matrix is also mentioned.

Dedicated to the memory of Harold Widom