New explicitly diagonalizable Hankel matrices related to the Stieltjes-Carlitz polynomials

TL;DR

This paper introduces four new explicitly diagonalizable Hankel matrices depending on a parameter, linked to Stieltjes-Carlitz polynomials, and explores their spectral properties using the commutator method.

Contribution

It presents novel explicit diagonalizations of Hankel matrices related to Stieltjes-Carlitz polynomials, expanding the class of structured matrices with known spectral solutions.

Findings

Four new parameter-dependent Hankel matrices are explicitly diagonalizable.

The spectral problem is solved using the commutator method.

Additional examples include weighted Hankel matrices.

Abstract

Four new examples of explicitly diagonalizable Hankel matrices depending on a parameter $k\in(0,1)$ are presented. The Hankel matrices are regarded as matrix operators on the Hilbert space $\ell^{2}(\mathbb{N}_{0})$ and the solution of the spectral problem is based on an application of the commutator method. Each of the Hankel matrices commutes with a Jacobi matrix which is related to a particular family of the Stieltjes-Carlitz polynomials. More examples of explicitly diagonalizable structured matrix operators are obtained when taking into account also weighted Hankel matrices.