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 in the bivariate difference field , where are known binary functions of , , and , are two algebraically independent transcendental elements, is a transformation that satisfies , , where . 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.

中文翻译：

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二元差分域中一阶差分方程的有理解

受卡尔算法的启发，我们考虑涉及满足二阶递推的序列的求和。这种求和的结构提供了一个代数框架，用于求解二元差分域中形式的差分方程，其中 、 和 是已知的二元函数，是两个代数独立的超越元素，是满足 、 、 其中 的变换。在此基础上，我们描述了在某些假设下找到二元差分域中这些方程的通用分母的算法。这将寻找此类方程的有理解的一般问题简化为寻找此类方程的多项式解的问题。