A summation framework is developed that enhances Karr's difference field approach. It covers not only indefinite nested sums and products in terms of transcendental extensions, but it can treat, e.g., nested products defined over roots of unity. The theory of the so-called $R\mathrm{\Pi }{\mathrm{\Sigma }}^{⁎}$-extensions is supplemented by algorithms that support the construction of such difference rings automatically and that assist in the task to tackle symbolic summation problems. Algorithms are presented that solve parameterized telescoping equations, and more generally parameterized first-order difference equations, in the given difference ring. As a consequence, one obtains algorithms for the summation paradigms of telescoping and Zeilberger's creative telescoping. With this difference ring theory one gets a rigorous summation machinery that has been applied to numerous challenging problems coming, e.g., from combinatorics and particle physics.

开发了一个求和框架，可以增强Karr的差异领域方法。它不仅涵盖先验扩展的不确定嵌套和和乘积，而且还可以处理例如在统一根上定义的嵌套乘积。所谓理论$\mathrm{[R}\mathrm{\Pi }{\mathrm{\Sigma }}^{⁎}$-extensions补充有算法，这些算法可自动支持此类差异环的构建，并有助于解决符号求和问题。给出了在给定的差分环中求解参数化伸缩方程，以及更普遍的参数化一阶差分方程的算法。结果，人们获得了用于伸缩和Zeilberger创造性伸缩的总和范例的算法。借助这种差分环理论，人们得到了一种严格的求和机制，该求和机制已应用于众多挑战性问题，例如组合论和粒子物理学。