Rational solutions of ordinary difference equations

Abstract

In this paper, we generalize the results of Feng and Gao [Feng, R., Gao, X.S., 2006. A polynomial time algorithm to find rational general solutions of first order autonomous ODEs. J. Symbolic Comput., 41(7), 735–762] to the case of difference equations. We construct two classes of ordinary difference equations ($O\Delta \mathrm{Es}$) whose solutions are exactly the univariate polynomial and rational functions respectively. On the basis of these $O\Delta \mathrm{Es}$ and the difference characteristic set method, we give a criterion for an $O\Delta E$ with any order and nonconstant coefficients to have a rational type general solution. For the first-order autonomous (constant coefficient) $O\Delta E$, we give a polynomial time algorithm for finding the polynomial solutions and an algorithm for finding the rational solutions for a given degree.