Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter

TL;DR

This paper demonstrates that multiple binomial sums and epsilon expansions of hypergeometric functions with half-integer parameters can be expressed using harmonic polylogarithms and related special functions, aiding calculations in quantum field theory.

Contribution

It establishes that these sums and expansions are representable in terms of known special functions, extending previous results to arbitrary weight and depth.

Findings

Multiple binomial sums are expressible via Remiddi-Vermaseren functions.

Epsilon expansions with half-integer parameters are representable using harmonic polylogarithms.

Results facilitate analytical calculations in Feynman diagram evaluations.

Abstract

We continue the study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we show the following results: Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth (see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions. Theorem B: The epsilon expansion of a hypergeometric function with one half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are ratios of polynomials. Some extra materials are available via the www at this http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html