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.


33C70 Other hypergeometric functions and integrals in several variables
33C65 Appell, Horn and Lauricella functions