Expansion around half-integer values, binomial sums and inverse binomial sums

TL;DR

The paper addresses the need to expand transcendental functions around rational, especially half-integer, values, motivated by massive particle calculations in quantum field theory. It extends existing nested-sum techniques by introducing roots of unity, sum refinements, and Gamma-function expansions to handle p/q shifts, and it generalizes these methods to binomial and inverse binomial sums. The work provides concrete algorithmic frameworks (Type A/B) and demonstrates their application to massive loop and phase-space integrals, yielding explicit expressions in terms of multiple polylogarithms and log-like functions. This approach enables systematic, symbolic epsilon expansions essential for higher-order corrections in particle physics analyses.

Abstract

I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around half-integer values. I present algorithms which are suitable for an implementation within a symbolic computer algebra system. The method is an extension of the technique of nested sums. The algorithms allow in addition the evaluation of binomial sums, inverse binomial sums and generalizations thereof.