## Abstract

The analytic properties of the Baxter-Wu solution for the Ising model with pure triplet interactions on the triangular lattice are investigated. In particular, it is shown that the free energy per spin of the model can be written in the explicit form $-(F/k_{B}T)$ = (2|J|/$k_{B}$$\dot{T}$) + ln $\Lambda (u)$, where [$\Lambda (u)$]$^{-1}$ = $\frac{1}{(1+u)^{2}}$$F_{1}$ [$\frac{1}{2}$, $\frac{1}{6}$; $\frac{4}{3}$; $\frac{16u(1-u)^{2}}{(1+u)^{4}}$],, and u = exp (-4|J|/$k_{\text{B}}$T). From this basic result and the group properties of the hypergeometric differential equation it is proved that the functions 1/$\Lambda (u)$ and $\Lambda (u)$ satisfy linear Fuchsian differential equations of second and fourth order respectively. Recurrence relations for the coefficients in the low temperature expansions of 1/$\Lambda (u)$ and $\Lambda (u)$ are derived, and the singular behaviour of these functions in the neighbourhood of the critical point u$_{\text{c}}$ = 3-2$\surd $2 is established. A detailed analysis of the critical behaviour of the free energy, internal energy and specific heat C$_{H=0}$ is also carried out. Finally, it is noted that the mathematical structure of the Baxter-Wu solution is very similar to that of the Onsager solution for the square lattice Ising model with pair interactions.