Recurrence relation approach for expansion and connection coefficients in series of classical discrete orthogonal polynomials

Abstract

The two formulae expressing explicitly the difference derivatives and the moments of discrete orthogonal polynomials {Pn(x): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of Pn(x) themselves are proved. Two other formulae for the expansion coefficients of general-order difference derivatives qf(x), and for the moments xqf(x), of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Hahn-Charlier, Hahn-Meixner and Hahn-Kravchuk are described.