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

Inspired by Karr's algorithm, we consider the summations involving a sequence satisfying a recurrence of order two. The structure of such summations provides an algebraic framework for solving the difference equations of form a sigma(g) + bg = f in the bivariate difference field (F (alpha, beta), sigma), where a, b, f is an element of F (alpha, beta) \ {0} are known binary functions of alpha, beta, and alpha, beta are two algebraically independent transcendental elements, sigma is a transformation that satisfies sigma(alpha) = beta, sigma(beta) = u alpha + v beta, where u, v not equal 0 is an element of F. Based on it, we then describe algorithms for finding the universal denominator for those equations in the bivariate difference field under certain assumptions. This reduces the general problem of finding the rational solutions of such equations to the problem of finding the polynomial solutions of such equations. (c) 2024 Elsevier Ltd. All rights reserved.