Lambert $W$-Function Branch Identities

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.

Added appendix by Letong Hong and Shengton Zhang finishing the proof of Theorem 3.3