Complex branches of a generalised Lambert $W$ function arising from $p,q$--binomial coefficients

TL;DR

This paper explores the complex branches of a generalized Lambert W function related to the $ ext{psi}(x)$-function, revealing new mathematical links and structures with potential physical applications.

Contribution

It provides a detailed analysis of the complex branches of a generalized Lambert W function and constructs Riemann surfaces, uncovering new connections to the classical Lambert W function.

Findings

Analysis of complex branches of the generalized Lambert W function

Construction of Riemann surfaces under various conditions

Identification of new links to the classical Lambert W function

Abstract

The $\psi(x)$-function, which solves the equation $x = \sinh(aw)e^w$ for $0<a<1$, has a natural connection to the renowned Lambert $W$ function and also physical relevance through its connection to the Lenz-Ising model of ferromagnetism. We give a detailed analysis of its complex branches and construct Riemann surfaces from these under various conditions of $a$, unveiling intriguing new links to the Lambert $W$ function.