Hypergeometric functions associated with the Grassmannian \(G_{3,6}\).

*(Russian)*Zbl 0667.33008I. M. Gel’fand [Dokl. Akad. Nauk SSSR 288, 14-18 (1986; Zbl 0645.33010)] has given the definition of general hypergeometric functions on the Grassman manifolds \(G_{k,n}\). The present paper is devoted to detailed description of the generalized hypergeometric functions on \(G_{3,6}\). These functions have five types (I-V) of non-degenerate singular points. The generalized hypergeometric functions on \(G_{3,6}\) regular in a neighborhood of any singular point, are represented in the form of power series. It is proved that in a neighborhood of any singular point of one of the types I-IV there is a unique regular generalized hypergeometric function. This function is given by a series of the hypergeometric type. There are several hypergeometric power series representations for the same generalized hypergeometric function. The number of these representations depends on the type of the singular point. It is shown that 2880 different hypergeometric power series correspond to fixed generalized hypergeometric functions on \(G_{3,6}\). It is proved that two linearly independent regular generalized hypergeometric functions correspond to a singular point of the type V. They are given by the series which are not of hypergeometric type.

There are six linearly independent generalized hypergeometric functions on \(G_{3,6}\). The authors construct all six linearly independent generalized hypergeometric functions (in form of power series), which are regular in some domain of \(G_{3,6}\). Further, the hypergeometric functions of one, of two, and of three variables are studied, which are restrictions of generalized hypergeometric functions onto some submanifolds of \(G_{3,6}\). Their relations to classical hypergeometric functions is considered. The authors give integral representations of generalized hypergeometric functions.

There are six linearly independent generalized hypergeometric functions on \(G_{3,6}\). The authors construct all six linearly independent generalized hypergeometric functions (in form of power series), which are regular in some domain of \(G_{3,6}\). Further, the hypergeometric functions of one, of two, and of three variables are studied, which are restrictions of generalized hypergeometric functions onto some submanifolds of \(G_{3,6}\). Their relations to classical hypergeometric functions is considered. The authors give integral representations of generalized hypergeometric functions.

Reviewer: A.Klimyk