Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials

Abstract

Starting from the second-order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials ∗pn∗, we find the analytical expressions of the expansion coefficients of any polynomial rm(x) and of the product rm(x)qj(x) in series of the set ∗pn∗. These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric polynomials. Here qj(x) denotes an arbitrary discrete hypergeometric polynomial of degree j. The particular cases in which ∗rm∗ corresponds to the non-orthogonal families ∗xm∗, the rising factorials or Pochhammer polynomials ∗(x)m∗ and the falling factorial or Stirling polynomials ∗x[m]∗ are considered in detail. The connection problem between discrete hypergeometric polynomials, which here corresponds to the product case with m = 0, is also studied and its complete solution for all the classical discrete orthogonal hypergeometric (CDOH) polynomials is given. Also, the inversion problems of CDOH polynomials associated to the three aforementioned nonorthogonal families are solved.