##
**On hypergeometric functions in several variables. I: New integral representations of Euler type.**
*(English)*
Zbl 0767.33009

The author defines a class of power series whose coefficients are products of shifted factorials \((\alpha;n)={{\Gamma(\alpha+n)} \over {\Gamma(n)}}\) and proves that every member of this class admits an Euler integral representation and that it satisfies a holonomic system. Appell- Lauricella’s Horn’s and Aomoto-Gel’fand’s hypergeometric functions are members of this class. In fact, defining the hypergeometric series in §1 he discusses convergence and integral representation in theorem 2, in the proof of which a crucial role is played by Kummer’s trick and the twisted cycle \(\Delta^ m(w)\) which is a higher dimensional version of classical double circuit and then establishes theorem 3 giving what may be termed as a better form of integral representation. In the second chapter of the paper applications and theorems 2 and 3 are given by obtaining new integral representations for Horn’s series \(G_ 3\), \(H_ 5-H_ 7\) and also by showing that \(F_ c\) defined in §1 admits an Euler integral representation which is a generalization of that for \(F_ 4\) by K. Aomoto [Group representations and systems of differential equations, Proc. Symp., Tokyo 1982, Adv. Stud. Pure Math. 4, 165-179 (1984; Zbl 0596.32015)] and that of \(F_ c\) due to P. I. Pastro [Bull. Sci. Math., II. Ser. 113, No.1, 119-124 (1989; Zbl 0668.33003)]. The integral obtained by the author is in the generalised case, a product of powers of linear and quadratic polynomials which is in contrast with the integral representation of Aomoto-Gel’fand hypergeometric series whose integral is a product of powers of linear polynomials only. The paper is concluded by making a remark on the duality of the Aomoto-Gel’fand hypergeometric functions found by I. M. Gel’fand and M. I. Graev and by presenting a system of differential equations satisfied by hypergeometric series, called the hypergeometric system and by giving estimates of the rank (the dimension of the solution space) of the system.

Reviewer: C.M.Joshi (Udaipur)

### MSC:

33C70 | Other hypergeometric functions and integrals in several variables |

33C65 | Appell, Horn and Lauricella functions |