On a multidimensional system of hypergeometric differential equations
type: Item
instance of: scholarly article
title: On a multidimensional system of hypergeometric differential equations
zbMATH Open document ID: 0920.34034
DOI: 10.1007/BF02672921
published in: Siberian Mathematical Journal
publication date: 1999-05-11T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34B30
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 65J99
zbMATH DE Number: 1286329
zbMATH Keywords: hypergeometric differential equations
zbMATH Keywords: Horn hypergeometric system
zbMATH Keywords: integral representation of a solution
zbMATH Keywords: solvability criterion
zbMATH Keywords: Ukawa connection constants
author: T. M. Sadykov
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02672921
OpenAlex ID: W2030589167
description: scientific article; zbMATH DE number 1286329