The birth and death processes are closely related to the orthogonal polynomials. The latter allows determining the stochastic matrix associated with these processes. Let us also note that these processes are stationary Markov processes whose state space is the nonnegative integers. Many authors treated the question of the existing relationship between the birth and death processes and the ortho...

The symmetric Al-Salam–Chihara polynomials for q > 1 are associated with an indeterminate moment problem. There is a self-adjoint second order difference operator on l(Z) to which these polynomials are eigenfunctions. We determine the spectral decomposition of this self-adjoint operator. This leads to a class of discrete orthogonality measures, which have been obtained previously by Christianse...

The symmetric Al-Salam and Chihara polynomials for q > 1 are associated with an indeterminate moment problem. There is a self-adjoint second order difference operator on l(Z) to which these polynomials are eigenfunctions. We determine the spectral decomposition of this self-adjoint operator. This leads to a class of discrete orthogonality measures, which have been obtained previously by Christi...