On diagonalizable quantum weighted Hankel matrices

TL;DR

This paper explicitly diagonalizes certain quantum weighted Hankel matrices using orthogonal polynomials, leading to new integral formulas and highlighting open problems in quantum matrix analysis.

Contribution

It introduces explicit diagonalizations of quantum weighted Hankel matrices via orthogonal polynomial techniques, expanding understanding of their spectral properties.

Findings

Diagonalization of Hankel matrices using Al-Salam-Chihara polynomials

Derivation of new integral formulas for quantum orthogonal polynomials

Identification of open problems related to quantum Hilbert 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.