  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.

Classical Analysis and ODEs