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

The $\psi (x)$-function, which solves the equation $x = \sinh (aw)e^w$ for $0 \lt a \lt 1$, 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 provide 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.