GG-functions and their relations to A-hypergeometric functions

In [1] I.M.Gelfand introduced the conception of hypergeometric functions, associated with the Grassmanian Gk,n of k-dimensional subspaces in C . The class of these functions includes many classical hypergeometric functions as particular cases. Series of consequent papers of I.M.Gelfand and his coathors ([2]–[6]) are devoted to the theory of these functions. In particular, the paper [4] is devoted to hypergeometric functions associated with the Grassmanian G3,6 . It should be mentioned that some well-known facts concerning classical hypergeometric functions were simply interpreted in these investigations. For example: 24 Kummer relations are known for the Gauss hypergeometric function F (a, b, c;x), which was proved to be connected with the Grassmanian G2,4 . These relations arise from the natural action of the permutation group S4 at G2,4 . The Appel function F1 is connected with the Grassmanian G2,5 . The existence of two integral representations for F1 by Euler integrals (one by single and one by double integrals) arises from the isomorphism of Grassmanians G2,5 and G3,5 . Further development of the theory of hypergeometric functions is associated with the paper of I.M.Gelfand, M.I.Graev and A.V.Zelevinsky [7]. In this paper general hypergeometric systems of equations were defined and their holonomity was proved. These general hypergeometric systems are called A-hypergeometric systems or GGZ-systems. Any A-hypergeometric system is defined by a set A = {ω, . . . , ω} of vectors of Z that linearly generate C, and by a vector β ∈ C. It consists of the