Meromorphic solutions of linear differential systems, Painlevé type functions

Abstract

Abstract. We consider the n × 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 Painlevé type functions. The fifth Painlevé function belongs to this class. Hence we obtain a new and simple proof that the fifth Painlevé function is meromorphic. (2000): 34M05, 34M55, 47B38. Mathematics subject classification

On the solutions of Knizhnik-Zamolodchikov system

“…Hence the assertion of the theorem is true for all integers ρ We note that (11) are true [7]. The theorem is proved.…”

Rationality of the Knizhnik-Zamolodchikov equation solution

“…Here can be useful the following necessary condition see [9]): Proposition 1.1 If ρ is integer and the fundamental solution of system (1.1), (1.2) is rational then all the eigenvalues of matrices…”

Explicit rational solutions of Knizhnik-Zamolodchikov equation

“…Proposition 1.1. (necessary condition, (see [4]) If the solution of system (1.1) has form (1.4) then m is an eigenvalue of ρa −1 .…”

“…Proposition 1.2. (necessary and sufficient condition, (see [4]) If the matrix system [(q + 1) 2 Calculation of the rational solution, general scheme…”