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

The paper investigates analytic coefficients in the $\varepsilon$-expansion of hypergeometric functions relevant to Feynman diagrams. It proves that multiple (inverse) binomial sums of arbitrary weight and depth can be expressed via Remiddi–Vermaseren functions, and that the all-order $\varepsilon$-expansion of a class of $_pF_{p-1}$ functions with a half-integer parameter reduces to harmonic polylogarithms with polynomial coefficients. By deriving differential equations for generating functions and employing strategic variable changes, the authors provide a constructive framework for both (inverse) binomial sums and hypergeometric expansions, including an algorithmic route to basis reduction. They also extend the approach to generalized sums via derivatives of hypergeometric functions, highlighting potential algebraic relations and open questions about completeness and connections to multiple zeta values, with practical implications for multi-loop diagram calculations.

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