## 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

### Citations:

Zbl 0767.33009; Zbl 0619.33006
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