Recurrences and explicit formulae for the expansion and connection coefficients in series of classical discrete orthogonal polynomials

Abstract

Two formulae expressing explicitly the difference derivatives and the moments of a discrete orthogonal polynomials {P n (x): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of P n (x) themselves are proved. Two other formulae for the expansion coefficients of a general-order difference derivatives ∇ q f(x), and for the moments x ℓ∇ q f(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 two families of Meixner, Kravchuk and Charlier, is described. Three analytical formulae for the connection coefficients between Hahn–Charlier, Hahn–Meixner and Hahn–Kravchuk are also developed.