Convergence in $\C$ of series for the Lambert $W$ Function

Abstract: We study some series expansions for the Lambert $W$ function. We show that known asymptotic series converge in both real and complex domains. We establish the precise domains of convergence and other properties of the series, including asymptotic expressions for the expansion coefficients. We introduce an new invariant transformation of the series. The transformation contains a parameter whose effect on the domain and rate of convergence is studied theoretically and numerically. We also give alternate representations of the expansion coefficients, which imply a number of combinatorial identities.