Treffer: ON THE RANK OF THE HYPERGEOMETRIC SYSTEM E(n+1, m+1; α): On the rank of the hypergeometric system \(E(n+1,m+1;\alpha)\)

Summary: In a previous paper [J. Math. Soc. Japan 45, No. 4, 645-669 (1993; Zbl 0799.33009)], we showed that the system of hypergeometric differential equations \(E(n+1, m+1; \alpha)\) admits \({m-1 \choose n}\) linearly independent hypergometric integrals as its solutions. However, logically speaking, the system may have another solution which is not written as a linear combination of the integrals. [\textit{I. M. Gel'fand}, \textit{A. V. Zelevinskij} and \textit{M. M. Kapranov}, Func. Anal. Appl. 23, No. 2, 94-106 (1989; Zbl 0787.33012)] has already given its solutions as power series and the number of its rank in view of the theory of \(D\)-modules and toral manifolds. However, in this paper we shall show that the rank of the system is equal to \({m-1 \choose n}\) in a direct and simple way. Of course a set of all linearly independent solutions of the system is given by the integrals. To prove the statement, we use essentially an identity of determinants of differential operators, which has the analogous form to the Capelli identity.