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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:247

Article

Balser, Werner¹  Roescheisen, Claudia¹ 

[1] Univ Ulm, Inst Angew Anal, D-89069 Ulm, Germany

关键词: Ordinary differential equations;    Power series solutions;    Generalized hypergeometric functions;   

DOI  :  10.1016/j.jde.2009.07.036

来源: Elsevier

【 摘 要 】

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. (C) 2009 Elsevier Inc. All rights reserved.

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