Solution of the pentagon equation for quantum $6j$-symbols with the help of multidimensional orthogonal polynomials

The pentagon relation is one of the non-linear relations for quantum $6j$-symbols. It is known that using the equation in a certain way, one can recursively find the value of any given $6j$-symbol. However, it is still very difficult to obtain analytical solutions that would describe the whole class of $6j$-symbols at once parametrically. We will explain that for the (quantum) group $sl(2)$ the pentagon equation can be rewritten as a recursive three-term relation on the orthogonal polynomial ($q$-)Racah, and thus any $6j$-symbol of this group can be expressed in terms of this orthogonal polynomial. We also present a number of considerations and our calculations showing that for groups of higher rank the pentagon relation should turn into counterparts of three-term relations for multidimensional orthogonal polynomials.