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Meixner polynomials of the second kind and quantum algebras representing su(1,1)

We show how Viennot's combinatorial theory of orthogonal polynomials may be used to generalize some recent results of Sukumar and Hodges on the matrix entries in powers of certain operators in a representation of su(1,1). Our results link these calculations to finding the moments and inverse polynomial coefficients of certain Laguerre polynomials and Meixner polynomials of the second kind. As an immediate consequence of results by Koelink, Groenevelt and Van Der Jeugt, for the related operators, substitutions into essentially the same Laguerre polynomials and Meixner polynomials of the second kind may be used to express their eigenvectors. Our combinatorial approach explains and generalizes this "coincidence".