IBIS: Inverse BInomial sum Solver

Abstract

In higher-loop calculations, Mellin-Barnes representations are used to simplify the denominators encountered in Feynman parameter integrals. The contour integrals of these representations yield sums over residues. We develop an efficient method for certain classes of sums involving inverse binomials. Our results are expressed in terms of so-called S-sums, where the dependence on the upper limit of the sum is analytic. This is accomplished by deriving several new recursion relations, obtained from telescoping series and repeated synchronization. We make our results available through IBIS ("Inverse BInomial sum Solver"), a FORM program to perform such inverse binomial sums. To highlight the efficiency of our code: sums up to weight 6 can be carried out in less than a second, and a comparison to the general-purpose SIGMA and EvaluateMultiSums packages is included. We show in examples how inverse binomial sums arise, though some instances are beyond the cases studied here. Our work thus provides a starting point for studying such sums and may open up new avenues in higher-loop calculations.

Comments: 28 pages including 1 appendix. Associated code available at: https://github.com/cbmarini/IBIS, v2: Updated examples and references