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Generalized Euler integrals and \(A\)-hypergeometric functions. (English) Zbl 0741.33011
In their previous works the authors have introduced a holonomic system of linear differential equations associated to a certain finite set \(A\subset \mathbb{Z}^ n\). They called this system the \(A\)-hypergeometric system, and its solutions the \(A\)-hypergeometric functions. The basis in the space of these functions consisting of series of hypergeometric type has been constructed. In the present paper the authors study integral expressions for these series. The main result is that for any \(A\)- hypergeometric system a complete set of its solutions can be represented as integrals of the form \[ \int\prod_ i P_ i(x_ 1,\dots,x_ k)^{\alpha_ i} x_ 1^{\beta_ 1}\dots x_ k^{\beta_ k} dx_ 1\dots dx_ k, \] where \(P_ i\) are polynomials; the integrals are considered as functions of the coefficients of \(P_ i\). The authors call these integrals generalized Euler integrals. They generalize the classical Euler integral for the Gauss hypergeometric function to the case of several variables.

33C70 Other hypergeometric functions and integrals in several variables
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