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

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.