On generalized Lambert function

Alexander Kreinin; Andrey Marchenko; Vladimir Vinogradov

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arXiv · 2504.07142

On generalized Lambert function

Alexander Kreinin·Andrey Marchenko·Vladimir Vinogradov

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Abstract

We consider a particular generalized Lambert function, $y(x)$, defined by the implicit equation $y^\beta = 1 - e^{-xy}$, with $x>0$ and $ \beta > 1$. Solutions to this equation can be found in terms of a certain continued exponential. Asymptotic and structural properties of a non-trivial solution, $y_\beta(x)$, 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.

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Alexander Kreinin, Andrey Marchenko, Vladimir Vinogradov. 2025-04-08. On generalized Lambert function. https://arxiv.org/abs/2504.07142

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