# Discovering and Proving Infinite Pochhammer Sum Identities

## TL;DR

This paper develops methods to transform complex nested Pochhammer sums into expressions involving well-known constants like pi, logs, and zeta values, using integral representations and harmonic polylogarithms, with implementation in HarmonicSums.

## Contribution

It introduces a novel approach to simplify infinite Pochhammer sums by expressing them through cyclotomic harmonic polylogarithms and constants, with algorithms implemented in HarmonicSums.

## Key findings

- Derivation of integral representations for nested sums.
- Expression of sums in terms of known constants.
- Implementation of methods in the HarmonicSums software.

## Abstract

We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. 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 or directly in terms of cyclotomic harmonic polylogarithms. Using substitutions, we express the root-valued iterated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants. The methods are implemented in the computer algebra package HarmonicSums.