Correlation functions of the Ising model with multisite interaction on the Husimi lattice

We consider a general spin-1/2 Ising model with multisite interaction on the Husimi lattice with the coordination number q and derive an analytical expression of correlation functions for stable fixed paints of the corresponding recurrence relation. We show that for q = 2 our model transforms to the two-state vertex model on the Bethe lattice with q = 3 and for the case q = 3, with only nearest neighbor interactions, we transform our model to the corresponding model on the Bethe lattice with q = 3, using the Yang-Baxter equations.