嵌套求和与超几何展开的一种新算法
A Novel Algorithm for Nested Summation and Hypergeometric Expansions
摘要：
考虑一类自变量相差一个符号整数的Z-和的乘积上的和。例如，这种和出现在高斯超几何函数围绕依赖于符号参数的整数指数的展开中。我们提出了一个伸缩算法，有效地将这些款项转化为广义多项式，Z-总和，分圆调和和通用值这个参数。通过十个循环计算双五阶积分，并通过维数正则化的$O(ε^6)$计算一族大量的自能图来说明该算法。我们还概述了该算法的一般伸缩策略，我们预计可以应用于其他类别的总和。
英文摘要：
We consider a class of sums over products of Z-sums whose arguments differ by a symbolic integer. Such sums appear, for instance, in the expansion of Gauss hypergeometric functions around integer indices that depend on a symbolic parameter. We present a telescopic algorithm for efficiently converting these sums into generalized polylogarithms, Z-sums, and cyclotomic harmonic sums for generic values of this parameter. This algorithm is illustrated by computing the double pentaladder integrals through ten loops, and a family of massive self-energy diagrams through $O(ε^6)$ in dimensional regularization. We also outline the general telescopic strategy of this algorithm, which we anticipate can be applied to other classes of sums.
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