Value of generalized hypergeometric function at unity

Abstract

Value of generalized hypergeometric function at a special point is calculated. More precisely, value of certain multiple integral over vanishing cycle (all arguments collapse to unity) is calculated. The answer is expressed in terms of $\Gamma$-functions. The constant is relevant to the part of $\rho$ in the Gindikin-Karpelevich formula for c-function of Harish-Chandra. Calculation is an adaptation of classical calculations of Gelfand and Naimark (1950) to the Heckman-Opdam hypergeometric functions in the case of root system of type $A_{n-1}$.