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With only a complete solution in dimension one and partially solved indimension two, the Lenz-Ising model of magnetism is one of the most studiedmodels in theoretical physics. An approach to solving this model in thehigh-dimensional case ($d>4$) is by modelling the magnetisation distributionwith $p,q$-binomial coefficients. The connection between the parameters $p,q$and the distribution peaks is obtained with a transition function $\omega$which generalises the mapping of Lambert $W$ function branches $W_0$ and$W_{-1}$ to each other. We give explicit formulas for the branches for specialcases. Furthermore, we find derivatives, integrals, parametrizations, seriesexpansions, and asymptotic behaviors.

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