The determinant of a hypergeometric period matrix and a generalization of Selberg's integral

In an earlier paper [D. Richards, Q. Zheng, Determinant formulas for multidimensional hypergeometric period matrices, Adv. in Appl. Math. 29 (2002) 137-151] on the determinants of certain period matrices, we formulated a conjecture about the determinant of a certain hypergeometric matrix. In this article, we establish this conjecture by constructing a system of linear equations in which that determinant is one of the variables. As a consequence, we obtain the value of an integral which generalizes the well-known multidimensional beta integral of A. Selberg [A. Selberg, Bemerkninger om et multipelt integral, Norsk. Mat. Tidsskr. 26 (1944) 71-78] and some hypergeometric determinant formulas of A. Varchenko [A. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. I, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 1206-1235, English translation, Math. USSR-Izv. 35 (1990) 543-571; A. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. II, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990) 146-158, English translation, Math. USSR-Izv. 36 (1991) 155-167].