Search of rational solutions to differential and difference systems by means of formal series

Abstract

Systems of linear ordinary differential and difference equations of arbitrary order with polynomial coefficients are considered. An algorithm for searching their rational solutions is suggested. The algorithm is based on the fact that these solutions are expanded into formal series that become polynomials after multiplication by a universal denominator. A combined algorithm relying on heuristics is also described. To reduce the amount of computation, at a certain moment, the latter selects between two algorithms, the new one and the standard algorithm based on the use of the universal denominator for changing unknowns and subsequent search for polynomial solutions.