On the explicit determination of certain solutions of periodic differential equations of higher order
type: Item
instance of: scholarly article
title: On the explicit determination of certain solutions of periodic differential equations of higher order
zbMATH Open document ID: 0855.30023
DOI: 10.1007/BF03322203
author: Anja Baesch
published in: Results in Mathematics
publication date: 1997-02-16T00:00:00Z
review text: In [Complex Variables, Theory Appl. 23, No. 1-2, 101-121 (1993; Zbl 0801.34006)], \textit{S. Bank} developed a method to test a linear differential equation \(y''+ A(z)y= 0\), \(A(z)\) a non-constant periodic entire function which is a rational function of \(e^z\), for the existence of solutions \(f\) satisfying \(\lambda(f)< \infty\), and to specify these solutions. In this paper, the method will be extended and simplified to cover the class of linear differential equations \[ y^{(k)}+ \sum^{k- 2}_{\nu= 1} A_\nu y^{(\nu)}+ A_0(z) y= 0,\tag{\(*\)} \] where \(A_1,\dots, A_{k- 2}\) are complex constants, \(k\geq 3\), and \(A_0(z)\) is a non-constant periodic entire function which is a rational function of \(e^z\). In fact, the idea of S. Bank to determine approximate square roots of a rational function will be extended to determine approximate \(k\)th roots. Then \(k^2\) linear differential equations will be constructed in such a way that (1) if none of these equations possesses a non-trivial polynomial solution, then \((*)\) possesses no non-trivial solution \(f\) such that \(\log^+ N(r, {1\over f})= o(r)\) and (2) whenever anyone of the \(k^2\) linear differential equations possesses a non-trivial polynomial equation, then this polynomial determines a solution \(f\) of \((*)\) such that the exponent of convergence for its zero-sequence \(\lambda(f)< \infty\).
reviewed by: Ilpo Laine
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 34M99
zbMATH DE Number: 872697
MaRDI profile type: Publication
cites work: On the explicit determination of certain solutions of periodic differential equations
cites work: Q3037791
cites work: On the frequency of zeros of solutions of second order linear differential equations
cites work: Q4019381
cites work: Oscillation theory for higher order linear differential equations with entire coefficients
cites work: Q3227756
full work available at URL: https://doi.org/10.1007/bf03322203
OpenAlex ID: W2094233747
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description: scientific article; zbMATH DE number 872697