EXPLICIT FORMULAS FOR COEFFICIENTS IN THE LAPPO-DANILEVSKY SOLUTION OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS

In the works of I.A. Lappo-Danilevsky, in particular, the solutions of a system of linear ordinary differential equations in the vicinity of an isolated pole of arbitrary finite order were investigated. For the fundamental matrix of solutions of such a system, a series was obtained that absolutely converges in the punctured (annular) neighborhood of the pole. At the same time, recurrent relations of a rather complex type were found for the numerical coefficients of the specified series, which do not depend on the type of the system of equations. In this paper, explicit formulas for these coefficients are obtained for the first time. As an example, the results obtained are used to find the analytical formula for the trace of the monodromy matrix of an arbitrary regular singular point of the specified system of equations.