Remarks on a theorem of Perron

TL;DR

This paper explores the convergence of generalized continued fractions related to solutions of linear differential equations with mild singularity conditions, focusing on the case of second-order equations and their logarithmic derivatives.

Contribution

It provides new insights into the convergence behavior of continued fractions associated with linear differential equations near singularities.

Findings

Convergence of generalized continued fractions is established under mild singularity conditions.

For second-order equations, the continued fractions relate to the logarithmic derivatives of solutions.

The results enhance understanding of the analytic structure of solutions near singularities.

Abstract

For a linear differential equation with a mild condition on its singularities, we discuss generalized continued fractions converging to expressions in its solutions and their derivatives. In the case of an order two linear differential equation, this is the logarithmic derivative of the holomorphic solution near a singularity.