On the solution of multidimensional hypergeometric differential equations using multiple residues (Q2716178)

The author considers the following differential equation: NEWLINE\[NEWLINE\begin{multlined} x_1,\dots,x_nP \left(x_1{\partial \over\partial x_1}, \dots, x_n {\partial \over\partial x_n}\right) y(x_1,\dots,x_n)-\\ -Q\left(x_1 {\partial \over \partial x_1},\dots, x_n{\partial \over\partial x_n}\right) y(x_1,\dots, x_n)=0, \end{multlined}NEWLINE\]NEWLINE where \(P,Q\) are polynomials represented as products of linear factors NEWLINE\[NEWLINEP(z_1,\dots,z_n)=t_1,\dots,t_n\prod^p_{i=1}\bigl(\langle A_i, z\rangle-c_i\bigr),NEWLINE\]NEWLINE NEWLINE\[NEWLINEQ(z_1,\dots,z_n)= \prod^q_{j=1} \bigl(\langle B_j,z \rangle- d_j\bigr),NEWLINE\]NEWLINE \(A_i,B_j, Z\in\mathbb{C}^n\).NEWLINENEWLINENEWLINEThis equation is one of the generalizations of the one-dimensional hypergeometric equation. Another approach is concerned with GKZ systems [\textit{I. M. Gelfand}, \textit{M. M. Kapranov} and \textit{A. V. Zelevinskij}, Adv. Math. 84, No. 2, 255-271 (1990; Zbl 0741.33011)]. The author gives the integral representation of a solution \(y\) of that equation in the form NEWLINE\[NEWLINEy(x_1, \dots, x_n)= \int_C\varphi(s_1, \dots, s_n)x_1^{s_1} \cdots x_n^{s_n} ds_1, \dots,ds_n,NEWLINE\]NEWLINE where \(C\) is some \(n\)-dimensional cycle and the function \(\varphi\) satisfies the functional equation NEWLINE\[NEWLINEP(s)\varphi (s)= Q(s+I) \varphi(s+I), \quad I=(1, \dots,1).NEWLINE\]NEWLINE Then this solution \(y\) is represented as a sum of some Laurent series.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00006].