New relations and identities for generalized hypergeometric coefficients

Generalized hypergeometric coefficients 〈pFq(a; b)|λ〉 enter into the problem of constructing matrix elements of tensor operators in the unitary groups and are the expansion coefficients of a multivariable symmetric function generalization pFq(a; b; z), z = (z1, z2,..., zt), of the Gauss hypergeometric function in terms of the Schur functions eλ(z), where λ = (λ1, λ2,..., λt) is an arbitrary partition. As befits their group-theoretic origin, identities for these generalized hypergeometric coefficients characteristically involve series summed over the Littlewood-Richardson numbers g(μνλ). Identities that may be interpreted as generalizations of the Bailey, Saalschütz,... identities are developed in this paper. Of particular interest is an identity which develops in a natural way a group-theoretically defined expansion over new inhomogeneous symmetric functions. While the relations obtained here are essential for the development of the properties of tensor operators, they are also of interest from the viewpoint of special functions.