Discovering and Proving Infinite Binomial Sums Identities

TL;DR

This paper develops a method to simplify infinite binomial and inverse binomial sums by expressing them through constants like powers of pi and logs, using integral representations and cyclotomic harmonic polylogarithms.

Contribution

It introduces a novel approach combining integral representations and cyclotomic harmonic polylogarithms to discover and prove identities for infinite binomial sums.

Findings

Expressed sums in terms of known constants

Derived integral representations using root-valued iterated integrals

Simplified sums via relations among cyclotomic harmonic polylogarithms

Abstract

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals. Using substitutions, we express the interated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants.