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We start by presenting a generalization of a discrete wave equation that isparticularly satisfied by the entries of the matrix coefficients of therefinement equation corresponding to the multiresolution analysis of Alpert.The entries are in fact functions of two discrete variables and they can beexpressed in terms of the Legendre polynomials. Next, we generalize thesefunctions to the case of the ultraspherical polynomials and show that these newfunctions obey two generalized eigenvalue problems in each of the two discretevariables, which constitute a generalized bispectral problem. At the end, wemake some connections to other problems.

Connection coefficients for ultraspherical polynomials with argument doubling and generalized bispectrality