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 mathcal{O}left({epsilon}^6right) in dimensional regularization. We also outline the general telescopic strategy of this algorithm, which we anticipate can be applied to other classes of sums.
Highlights
Generalized polylogarithms can in principle be exploited with the use of the coaction [5, 6], as applied in [7,8,9,10]
For generic values of N, these sums evaluate to cyclotomic harmonic sums, while in the N → ∞ limit they reduce to Z-sums
This algorithm allows us, in particular, to resum the expansion coefficients of Gauss hypergeometric functions in cases where this expansion is around integer indices that depend on a symbolic integer parameter α
Summary
We give just a brief review of Z-sums, introducing notation and recalling the properties that will prove useful in later sections. The sums one encounters often have different lower summation bounds than allowed in (2.2); in particular, we will see below that expansions of gamma functions that appear in the denominator of hypergeometric functions more naturally give rise to S-sums, which are closely related to Z-sums [16]. Which separates out the contribution to these sums in which pairs of summation indices are equal; this leaves summation bounds that fit the definition of Z-sums, at the cost of introducing a sum of lower depth (making it clear that this recursion terminates). These sums appear naturally in the expansion of the gamma function and its reciprocal, and in the expansion of hypergeometric functions
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