Probability Measures on Dual Objects to Compact Symmetric Spaces and Hypergeometric Identities

We derive, in several different ways, combinatorial identities which are multidimensional analogs of classical Dougall's formula for a bilateral hypergeometric series of the type ${}_2H_2$. These identities have a representation-theoretic meaning. They make it possible to construct concrete examples of spherical functions on inductive limits of symmetric spaces. These spherical functions are of interest to harmonic analysis.