Growth of analytic solutions of linear differential equations with analytic coefficients near a finite singular point

Karima Hamani and Meryem Chetti – GJM, Volume 10, Issue 1 (2025), 25-33.

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• juillet 29, 2025
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In this paper, we investigate the growth of analytic solutions of the linear differential equation

f^{(k)}+A_{k-1}(z)\exp\Big\lbrace \frac{a_{k-1}}{(z_{0}-z)^{n}}\Big\rbrace f^{(k-1)}+\cdots+A_{0}(z)\exp\Big\lbrace \frac{a_{0}}{(z_{0}-z)^{n}}\Big\rbrace f=0,

where n \in \mathbb{N}-\lbrace 0 \rbrace, k\geq 2 is an integer and A_{j}(z)(j=0, \ldots,k-1) are analytic functions in the closed complex plane except a singular point z_{0} and a_{j}(j=0, \ldots,k-1) are complex numbers. Under some conditions, we prove that these solutions are of infinite order and their hyper-order is equal to $n$. We also consider the nonhomogeneous linear differential equations.

Milestones:

Received: January 7, 2025
Accepted: April 18, 2025
Revised: April 20, 2025
Published: July 25, 2025

Authors:

Karima Hamani and Meryem Chetti
Department of Mathematics,
Laboratory of Pure and Applied Mathematics,
University of Mostaganem (UMAB),
B. P. 227 Mostaganem, Algeria

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