On generalized Lambert function

2025/04/08 by Kreinin, Alexander, Marchenko, Andrey, Vinogradov, Vladimir
#60E05 #60E07 #60F05 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2504.07142

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.

Citations

• Ten Lectures on the Moment Problem
• Tornheim-like series, harmonic numbers and zeta values
• Taylor series for generalized Lambert W functions
• On the generalization of the Lambert W function with applications in theoretical physics
• General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function
• Harmonic Double Series
• Euler Sums and Contour Integral Representations

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