New Explicitly Diagonalizable Hankel Matrices Related to the Stieltjes–Carlitz Polynomials

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.