Explicit Formulas for the Coefficients in the Lappo-DanilevskySolutionof Linear Ordinary Differential Equations

I. A. Lappo-Danilevsky in particular studied the solutions of a system of linear ordinary differential equations in a neighborhood of an isolated pole of arbitrary finite order. For the fundamental solution matrix of such a system, a series absolutely convergent in a punctured (annular) neighborhood of the pole was obtained; for the numerical coefficients of this series, which are independent of the type of the system of equations, recursion relations of a rather involved form were found. The present paper is the first to obtain closed-form expressions for these coefficients. By way of example, the results are used to find the trace of the monodromy matrix of an arbitrary regular singular point (first-order pole) of the system of equations in question in the form of a series that is an entire function of the entries of the constant matrix.