Implementation of an Algorithm for Finding Analyticity Conditions for a Solution of a Difference Equation

Abstract

A linear difference operator L with polynomial coefficients and a function F(x) satisfying the equation LF(x) = 0 are considered. The function is assumed to be analytic in the interval (–σ, d), where σ > 0. In the paper, an implementation of an algorithm suggested by S.A. Abramov and M. van Hoeij for finding conditions that guarantee analyticity of F(x) on the entire real axis $\mathbb{R}$ is presented. The analyticity conditions are linear relations for values of F(x) and its derivatives at a given point belonging to the half-interval [0, d). A procedure for computing values of F(x) and its derivatives up to a prescribed order at a given point x ₀ ∈ $\mathbb{R}$ is also implemented. Examples illustrating the program operation are presented.