A central connection problem for a normal system of linear differential equations

Kohno, Mitsuhiko; Yokoyama, Toshiaki

doi:10.32917/hmj/1206133037

Cited by 4 publications

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“…In general, however, the solutions of this system are believed to be "new" higher transcendental functions. Their global behaviour as well as their relation via Laplace transform to the confluent hypergeometric system (see below) have also been investigated by other authors, among them Balser, Jurkat, and Lutz [3,4], M. Kohno [7][8][9][10], Kohno and Yokoyama [11], Okubo and Takano [13], Okubo, Takano, and Yoshida [14], and R. Schäfke [16][17][18].…”

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confidence: 98%

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

Balser

Röscheisen

2009

Journal of Differential Equations

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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.

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confidence: 98%

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

Balser

Röscheisen

2009

Journal of Differential Equations

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Solutions of Systems of Linear Ordinary Differential Equations

Balser

Röscheisen

Steiner

et al. 2009

Mathematical Analysis of Evolution, Information, and Complexity

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On the central connection problem for equations with an irregular singular point of a single level

Balser

1997

Journal of Dynamical and Control Systems

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ABSTRACT. We study the central connection.problem for linear systems of equations with two singularities: one at the origin which is assumed to be regular-singular, and another one at infinity having a formal fundamental solution of only one level (in the Newton polygon). INTRODUCTIONThe so-called centra/connection problem (CCP for short) has been investigated, in various degrees of generality, by many authors. Without claim of completeness, we have included the relevant papers in the list of references (for articles older than 1989 we used a literature search which A. Mehl [21], then a student in Ulm, made for his graduation paper).In particular, M. Kohno [14]-[18] studied the CCP for certain equations and obtained explicit formulas for the central connection coemcients. Here, we shall not only treat a case which is (slightly) more general than that considered by Kohno, but also follow a more direct approach to the problem (although the basic idea is the same): we consider an n-dimensional system of linear ordinary differential equations having a regular singularity at the origin and, as the only other singular point, an irregular singularity at infinity. For the time being, we restrict ourselves to the cases where the Newton polygon infinity only has one slope, or in other words, the formal fundamental solution is ~-summable for a single level ~ (see, e.g., [4] for the definition of ~-summability, resp., multisummability). (This assumption is, however, not the same as saying that we have all distinct eingenvalues at infinity, so the formal monodromy matrix is allowed to be nondiagonalizable.1991 Mathematics Subject Classifica~or~ 34A25, 44A10.

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A central connection problem for a normal system of linear differential equations

Cited by 4 publications

References 3 publications

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

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

Solutions of Systems of Linear Ordinary Differential Equations

On the central connection problem for equations with an irregular singular point of a single level

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