Solving the hypergeometric system of Okubo type in terms of a certain generalized hypergeometric function

Abstract

We introduce one scalar function f of a complex variable and finitely many parameters, which allows to represent all solutions of the so-called hypergeometric system of Okubo type under the assumption that one of the two coefficient matrices has all distinct eigenvalues. In the simplest non-trivial situation, f is equal to the hypergeometric function, while in other more complicated cases it is related, but not equal, to the generalized hypergeometric functions. In general, however, this function appears to be a new higher transcendental one. The coefficients of the power series of f about the origin can be explicitly given in terms of a generalized version of the classical Pochhammer symbol, involving two square matrices that in general do not commute. The function can also be characterized by a Volterra integral equation, whose kernel is expressed in terms of the solutions of another hypergeometric system of lower dimension.