Lambert $W$-Function Branch Identities

TL;DR

This paper explores the Lambert W-function branches, providing numerous exact identities, convergent series, and introducing a new family of polynomials of potential independent interest.

Contribution

It offers a comprehensive set of identities and series for Lambert W-branches and introduces a novel family of polynomials.

Findings

Exact identities involving Lambert W-branches

Convergent series for all branches

Introduction of a new polynomial family

Abstract

After defining in detail the Lambert $W$-function branches, we give a large number of exact identities involving (infinite) symmetric functions of these branches, as well as geometrically convergent series for all the branches. In doing so, we introduce a family of polynomials which may be of independent interest.