Rational solutions of ordinary difference equations

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 () whose solutions are exactly the univariate polynomial and rational functions respectively. On the basis of these and the difference characteristic set method, we give a criterion for an with any order and nonconstant coefficients to have a rational type general solution. For the first-order autonomous (constant coefficient) , we give a polynomial time algorithm for finding the polynomial solutions and an algorithm for finding the rational solutions for a given degree.