Algorithms for Construction of Recurrence Relations for the Coefficients of the Fourier Series Expansions with Respect to Classical Discrete Orthogonal Polynomials

Abstract

A new formula expressing explicitly the integrals, antidifference, of discrete orthogonal polynomials $$\{P_{n}(x):$$ Hahn, Meixner, Kravchuk, and Charlier $$\}$$ of any degree in terms of $$P_{n}(x)$$ themselves are proved. Other formulae for the expansion coefficients of general-order difference integrations $$\nabla ^{-s}f(x),\,\Delta ^{-s}f(x),$$ $$\nabla ^{-s}[x^{\ell }\nabla ^{q}f(x)]\,$$ and $$\Delta ^{-s}[x^{\ell }\Delta ^{q}f(x)],$$ 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.