Rational solutions of ordinary difference equations

  • Authors:

  • Ruyong Feng;Xiao-Shan Gao;Zhenyu Huang

  • Affiliations:

  • Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Academia Sinica, Beijing 100190, China;Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Academia Sinica, Beijing 100190, China;Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Academia Sinica, Beijing 100190, China

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2008

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