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Abstract <p> I. A. Lappo-Danilevsky in particular studied the solutions of a system of linear ordinarydifferential equations in a neighborhood of an isolated pole of arbitrary finite order. For thefundamental 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 ratherinvolved form were found. The present paper is the first to obtain closed-form expressions forthese coefficients. By way of example, the results are used to find the trace of the monodromymatrix of an arbitrary regular singular point (first-order pole) of the system of equations inquestion in the form of a series that is an entire function of the entries of the constant matrix.</p>

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Explicit Formulas for the Coefficients in the Lappo-Danilevsky Solution of Linear Ordinary Differential Equations

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • A. A. Golubkov

摘要

Abstract

I. A. Lappo-Danilevsky in particular studied the solutions of a system of linear ordinarydifferential equations in a neighborhood of an isolated pole of arbitrary finite order. For thefundamental 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 ratherinvolved form were found. The present paper is the first to obtain closed-form expressions forthese coefficients. By way of example, the results are used to find the trace of the monodromymatrix of an arbitrary regular singular point (first-order pole) of the system of equations inquestion in the form of a series that is an entire function of the entries of the constant matrix.