Orthogonal polynomials, $6j$-symbols and statistical weights of SOS models

A simple diagrammatic method which allows to connect Boltzmann weights of the vertex models of statistical mechanics with those of SOS models is described. The analogy with the computation of the $6j-$symbols is pointed out. The construction of statistical weights heavily relies on the realization of the $SU(2)$ group on the space of functions of one variable. The closed-form answer for some particular cases is obtained. It is shown, that in the general case the statistical weight of SOS model, as well as $6j-$symbol, can be presented as the scalar product of two polynomials of certain type.