Taylor series for generalized Lambert W functions

TL;DR

This paper extends the Lambert W function to a more general class involving products of linear functions and exponentials, providing Taylor series coefficients via hypergeometric functions and proposing a conjecture on convergence radius.

Contribution

It introduces a new generalized Lambert W function, describes its Taylor series coefficients using multivariable hypergeometric functions, and conjectures the series' radius of convergence.

Findings

Coefficients of the series are expressed with hypergeometric functions.

A conjecture on the radius of convergence is proposed and roughly proved.

The generalization has applications in delay differential equations and physics.

Abstract

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.