On a generalised Lambert W branch transition function arising from p,q-binomial coefficients

With only a complete solution in dimension one and partially solved in dimension two, the Lenz-Ising model of magnetism is one of the most studied models in theoretical physics. An approach to solving this model in the high-dimensional case (d>4) is by modelling the magnetisation distribution with p,q-binomial coefficients. The connection between the parameters p,q and the distribution peaks is obtained with a transition function ω which generalises the mapping of Lambert W function branches W0 and W−1 to each other. We give explicit formulas for the branches for special cases. Furthermore, we find derivatives, integrals, parametrizations, series expansions, and asymptotic behaviours.