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Meromorphic Solutions of Linear Differential Systems, Painleve type functions

Lev Sakhnovich
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Abstract

We consider the $n{\times}n$ matrix linear differential systems in the complex plane. We find necessary and sufficient conditions under which these systems have meromorphic fundamental solutions. Using the operator identity method we construct a set of systems which have meromorphic solutions. We prove that the well known operator with the sine kernel generates a class of meromorphic Painleve type functions. The fifth Painleve function belongs to this class. Hence we obtain a new and simple proof that the fifth Painleve function is meromorphic.

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Lev Sakhnovich. 2006-07-21. Meromorphic Solutions of Linear Differential Systems, Painleve type functions. https://arxiv.org/abs/math/0607555

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