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

Abstract

We continue the study of the construction of analytical coefficients of the ?-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 where k = ?1, Sa(j) is a harmonic series, Sa(j) = ?jk = 1?1/ka, and c is any integer number are expressible in terms of Remiddi-Vermaseren functions; Theorem B: The hypergeometric functions are expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are ratios of polynomials.