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On hypergeometric functions in several variables. II: The Wronskian of the hypergeometric functions of type \((n+1,m+1)\). (English) Zbl 0799.33009

[For part I see the author in ibid. 18, No. 1, 25-74 (1992; Zbl 0767.33009.]
Although the fact that the rank of hypergeometric systems \(E(n + 1, m + 1; \lambda)\) is equal to \({m - 1 \choose n}\), where \(\lambda = (\lambda_ 1, \lambda_ 2, \dots, \lambda_{m + 1})\) are complex parameters, is fundamental in studying hypergeometric functions but no explicit statement with rigorous proof is known. For a particular case \(E(3,6; \lambda)\) [K. Matsumoto, T. Sasaki and M. Yoshida, Int. J. Math. 3, 1-164 (1992)] gave explicitly six solutions of the form \(y'\sum_{n\in z\geq 0} A(n)y^ n\), where \(y = (y_ 1, y_ 2, y_ 3, y_ 4)\) and \(p = (p_ 1, p_ 2, p_ 3, p_ 4)\); and I. M. Gel’fand and M. I. Graev [Sov. Math. Dokl. 34, 9-13 (1987; Zbl 0619.33006)] gave six solutions of \(E(3,6)\), but conditions under which the solutions are linearly independent are not given. In the present paper the author discusses this point in detail and proves that this is so, under a very simple nonintegral condition \(\lambda_ j \in \mathbb{C} - \mathbb{Z}\) \((1 \leq j \leq m)\) \((\mathbb{C}\) represents the set of all complex numbers and \(\mathbb{Z}\) represents the set of integers) using the theory of twisted rational de Rham cohomologies and twisted homologies.

MSC:

33C70 Other hypergeometric functions and integrals in several variables
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