Contiguity Relations of Aomoto–Gel’fand Hypergeometric Functions and Applications to Appell’s System ${}_3 F_2 $ and Goursat’s System ${}_3 F_2 $

Abstract

Aomoto–Gel’fand hypergeometric functions $\Phi (Z,\alpha )$ [K. Aomoto, Sci. Papers, College of Arts and Sciences, University of Tokyo, 27 (1977), pp. 49–61], [I. M. Gel’fand, Soviet Math. Dokl., 33 (1986), pp. 573–577] are functions of z defined on the Grassmannian $G_{k,n} $, the set of k-dimensional subspaces of an n-dimensional linear space, and with complex parameters $(\alpha )$. Such a class of functions contains certain classical hypergeometric functions (HGF), such as the HGF of Gauss, the generalized HGF ${}_{p + 1} F_p $, and Appell’s HGF’s $F_1 $, $F_2 $, $F_3 $. On the other hand, W. Miller [J. Math. Phys.,13 (1972), pp. 1393–1399; SIAM J. Appl. Math., 25 (1973), pp. 226–235; SIAM J. Math. Anal., 3 (1972), pp. 31–44] has given contiguity relations for several HGF’s, including the HGF’s mentioned above, and has shown the Lie-algebraic structure of the equations satisfied by these functions. This paper first presents a principle of obtaining contiguity relations for Aomoto–Gel’fand HGF’s and clarifies the Lie-algebraic structure among them. The contiguity relations for Lauricella’s HGF $F_D$ are known easily from this principle. The second part of this paper is an application to get complete tables of contiguity relations for $F_3$ and ${}_3 F_2 $, which complement the tables for these functions given by Miller in the papers cited above.