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

Per Åhag, Rafał Czyż, Per-Håkan Lundow

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…