On Hankel matrices commuting with Jacobi matrices from the Askey scheme

TL;DR

This paper characterizes all Hankel matrices that commute with Jacobi matrices from the Askey scheme, revealing that the generalized Hilbert matrix is uniquely diagonalizable via the commutator method among such matrices.

Contribution

It provides a complete characterization of Hankel matrices commuting with Jacobi matrices from the Askey scheme, highlighting the uniqueness of the generalized Hilbert matrix.

Findings

The generalized Hilbert matrix is the only prominent infinite-rank Hankel matrix diagonalizable by the commutator method with these Jacobi matrices.

A full classification of commuting Hankel matrices with Jacobi matrices from the Askey scheme is achieved.

The result links the structure of Hankel matrices to hypergeometric orthogonal polynomials in the Askey scheme.

Abstract

A complete characterization is provided of Hankel matrices commuting with Jacobi matrices which correspond to hypergeometric orthogonal polynomials from the Askey scheme. It follows, as the main result of the paper, that the generalized Hilbert matrix is the only prominent infinite-rank Hankel matrix which, if regarded as an operator on $\ell^{2}(\mathbb{N}_{0})$, is diagonalizable by application of the commutator method with Jacobi matrices from the mentioned families.