# zbMATH — the first resource for mathematics

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.)
Full Text: