Abstract
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 ℓ2(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 ℓ2-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.
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Christiansen, J., Koelink, E. Self-Adjoint Difference Operators and Symmetric Al-Salam–Chihara Polynomials. Constr Approx 28, 199–218 (2008). https://doi.org/10.1007/s00365-007-0677-x
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DOI: https://doi.org/10.1007/s00365-007-0677-x