A parafermionic hypergeometric function and supersymmetric 6j-symbols

Abstract

We study properties of a parafermionic generalization of the hyperbolic hypergeometric function appearing as the most important part in the fusion matrix for Liouville field theory and the Racah-Wigner symbols for the Faddeev modular double. We show that this generalized hypergeometric function is a limiting form of the rarefied elliptic hypergeometric function V(r) and derive its transformation properties and a mixed difference-recurrence equation satisfied by it. At the intermediate level we describe symmetries of a more general rarefied hyperbolic hypergeometric function. An important r=2 case corresponds to the supersymmetric hypergeometric function given by the integral appearing in the fusion matrix of N=1 super Liouville field theory and the Racah-Wigner symbols of the quantum algebra Uq(osp(1|2)). We indicate relations to the standard Regge symmetry and prove some previous conjectures for the supersymmetric Racah-Wigner symbols by establishing their different parametrizations.