Self-adjoint difference operators and symmetric Al-Salam-Chihara polynomials

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(2)(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 Christiansen and Ismail using a different method, and we give an explicit orthogonal basis for the corresponding weighted l(2)-space. In particular, the orthocomplement of the polynomials is described explicitly. Taking a limit we obtain all the N-extremal solutions to the q(-1)-Hermite moment problem, a result originally obtained by Ismail and Masson in a different way. Some applications of the results are discussed.