符号求和的差环理论
A Difference Ring Theory for Symbolic Summation
摘要：
一个求和框架的开发，提高卡尔的差场方法。它不仅涵盖了超越扩张方面的不定套和和积，但它可以处理，例如，在单位根上定义的嵌套乘积。理论的所谓的$RΠΣ^*$ -扩展是补充算法，支持建设这样的差环自动和协助的任务，以解决符号求和问题。算法，解决参数化的伸缩方程，更一般的参数化的一阶差分方程，在给定的差环。因此，一个获得算法的总和范例的伸缩和Zeilberger的创造性伸缩。有了这个差环理论，人们得到了一个严格的求和机制，它已被应用于许多具有挑战性的问题，例如，从组合学和粒子物理学。
英文摘要：
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 $RΠΣ^*$-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 obtains a rigorous summation machinery that has been applied to numerous challenging problems coming, e.g., from combinatorics and particle physics.
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