On a multidimensional system of hypergeometric differential equations
The author studies a multidimensional system of hypergeometric differential equations. Such systems arise in superstring theory while studying the Ukawa connection constants. As a multidimensional analog of the generalized hypergeometric differential equation, the following Horn hypergeometric system is considered: \[ Q_i\left(x\frac{\partial}{\partial x}\right)x_i^{-1}y(x_1,\dots ,x_n) = P_i\left(x\frac{\partial}{\partial x}\right)y(x_1,\dots ,x_n), \qquad i=1,\dots , n, \] where \(P_i\) and \(Q_i\) are polynomials. The author exhibits an integral representation for solutions to the system and writes down some system of difference equations whose fulfillment is a sufficient condition for the integral in question to satisfy the Horn system of equations. The main result of the article which contains a solvability criterion for the corresponding system of difference equations reads: A necessary and sufficient condition for this criterion to be valid is an agreement condition on the polynomials \(P_i\) and \(Q_i\). Moreover, if a solution to the system of difference equations exists then it is uniquely determined up to a factor satisfying the periodicity condition \(\varphi (s+e_i) \equiv \varphi (s)\) for all \(i=1,\dots ,n\). The author exposes a method for representing a solution to the Horn system in the form of a multiple series in the case of simple singularities. In the Gelfand theory, simple singularities correspond to the so-called nonresonance case.
scientific article; zbMATH DE number 1286329