Polynomial solutions to the first order difference equations in the bivariate difference field

Abstract

The bivariate difference field provides an algebraic framework to study sequences satisfying a recurrence of order two, and it can be used to transform summations involving such sequences into first order difference equations over the bivariate difference field. Based on this, we present an algorithm for finding a series of polynomial solutions of such equations in the bivariate difference field, and show an upper bound on the degree of any possible polynomial solutions, which in turn is sufficient to compute all polynomial solutions by using the method of undetermined coefficients.

Keywords:

• Difference equations
• polynomial solutions
• bivariate difference field
• recurrence relation

2020 Mathematics Subject Classifications:

• 33F10
• 12H10
• 11B37

Acknowledgments

The author would like to thank Qing-Hu Hou for his advice and comments connected with the subject of this paper, and two referees for their careful reading and constructive suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).