On generalized Lambert function

TL;DR

This work generalizes the Lambert W framework by introducing a generalized Lambert function defined implicitly by $y^\beta = 1 - e^{-xy}$ with $x>0$ and $\beta>1$, and explores its rich connections to extinction probabilities in Galton-Watson processes. It establishes the inverse representation $y_\beta(x)$ of $x_\beta(y) = -\log(1-y^\beta)/y$, proves monotonicity and a continued-exponential representation, and derives two-sided asymptotic bounds and contraction properties near the fixed point. The paper develops a probabilistic interpretation by showing $y_\beta(x)$ is the cdf of an absolutely continuous r.v. $\xi_\beta$, and provides closed-form moment expressions, a uniform random-variable representation, and connections to Tornheim sums and zeta-values. It also offers practical numerical methods, including efficient asymptotic approximations and iteration counts, and analyzes the moment problem for $\xi_\beta$ to establish determinacy. Collectively, the results give a comprehensive analytic and numerical treatment of a generalized Lambert function with applications to branching processes and stochastic modeling.

Abstract

We consider a particular generalized Lambert function, $y(x)$, defined by the implicit equation $y^β= 1 - e^{-xy}$, with $x>0$ and $ β> 1$. Solutions to this equation can be found in terms of a certain continued exponential. Asymptotic and structural properties of a non-trivial solution, $y_β(x)$, and its connection to the extinction probability of related branching processes are discussed. We demonstrate that this function constitutes a cumulative distribution function of a previously unknown non-negative absolutely continuous random variable.

Figures (12)

• Figure 1: The function ${\mathcal{F}}(x)$.
• Figure 2: Graph of the generalized Lambert function.
• Figure 3: Iterations of the mapping $\widehat{{\mathcal{W}}}$.
• Figure 4: Surface of the function $y_\beta(x)$.
• Figure 5: Graph of the function $z(t)$.

Theorems & Definitions (51)

• Proposition 2.1
• proof
• Definition 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Lemma 3.5
• proof
• Proposition 3.6
• proof