A Novel Algorithm for Nested Summation and Hypergeometric Expansions

Abstract

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 $O({\epsilon }^6)$ in dimensional regularization. We also outline the general telescopic strategy of this algorithm, which we anticipate can be applied to other classes of sums.

arXiv comment36 pages, 2 figures; v2: references added, typos corrected, improved introduction and comparison with existing methods, matches published version

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• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
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• Hexagon Bootstrap in the Double Scaling Limit