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On the system of differential equations associated with a quadric and hyperplanes. (English) Zbl 0888.33009
The authors consider the integral \(\int q^{a_{n+1}-1}\prod _{i=1}^n l_i^{a_i-1}\omega _k\), where \(l_i\) are linear forms, \(q\) is a quadratic form on the projective space \({\mathbf P}^{k-1}(t)\) of dimension \(k-1\) and \(\omega _k=\sum _{i=1}^k (-1)^{i-1}t^idt^1\wedge \cdots \wedge dt^{i-1}\wedge dt^{i+1}\wedge \cdots \wedge dt^k\). Here \(t=(t^1,\cdots ,t^k)\) denotes the homogeneous coordinates and \(a_1+\cdots +a_n+2a_{n+1}=n-k+2\). The integral is a function of the coefficients of \(l_i\) and \(q\). The authors derive the system of differential equations (denoted by \(Q(k,n)\)) associated with the integral. The system is defined on the coarse configuration space, denoted by \(X\), and the group \(G= (GL_k\times ({\mathbf C}^*)^{n+1})/{\mathbf C}^*\) acts on this space. The system is of the type of the system for Gel’fand hypergeometric functions and has a symmetry called the contiguity relations. Then the authors derive the system induced on the quotient space \(GL_k\backslash X\) and pay a special attention to the system \(Q(3,4)\). An explicit solution of the system \(Q(3,4)\) is given by a power series. The relation of this solution to classical hypergeometric functions is discussed.
Reviewer: A.Klimyk (Kyïv)

MSC:
33C70 Other hypergeometric functions and integrals in several variables
35C15 Integral representations of solutions to PDEs
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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