Taylor series for generalized Lambert W functions

The Lambert W function gives the solutions of a simple exponential polynomial. The generalized Lambert W function was defined by Mez\"{o} and Baricz, and has found applications in delay differential equations and physics. In this article we describe an even more general function, the inverse of a product of powers of linear functions and one exponential term. We show that the coefficients of the Taylor series for these functions can be described by multivariable hypergeometric functions of the parameters. We also present a surprising conjecture for the radius of convergence of the Taylor series, with a rough proof.