Linear Differential Equations with Analytic Coefficients Having the Same Order Near a Singular Point

Abstract

In this paper, we investigate the growth of solutions of the differential equation: $$\begin{aligned} f^{\left( k\right) }+A_{k-1}\left( z\right) \exp \left\{ \frac{a_{k-1}}{ \left( z_{0}-z\right) ^{n}}\right\} f^{\left( k-1\right) }+\cdots +A_{0}\left( z\right) \exp \left\{ \frac{a_{0}}{\left( z_{0}-z\right) ^{n}}\right\} f=0, \end{aligned}$$ where $$A_{j}\left( z\right) $$ are analytic functions in the closed complex plane except at $$z_{0}$$ and $$a_{j}\left( j=0,\ldots ,k-1\right) $$ are distinct complex numbers. Other linked cases have been studied.