Linear differential equations with rational coefficients
type: Item
instance of: scholarly article
title: Linear differential equations with rational coefficients
zbMATH Open document ID: 0876.34002
published in: Journal of the Mathematical Society of Japan
publication date: 1997-09-17T00:00:00Z
review text: This article considers meromorphic solutions of the linear differential equation \[ w^{n}+r_{n-2}w^{n-2}+\cdots + r_{0}w=0\tag{1} \] with rational coefficients. As well-known, if the coefficients are polynomials, then all solutions of (1) are entire functions such that almost all of their zeros are attracted to a finite number of radii. Moreover, \textit{F. Brüggemann} [Proc. Am. Math. Soc. 113, No. 2, 371-379 (1991; Zbl 0733.30024)] and \textit{N. Steinmetz} [Analysis 11, No. 2/3, 119-128 (1991; Zbl 0744.34014)] proved the Frank-Wittich conjecture that the coefficient polynomials reduce to constants if (1) has a fundamental set of solutions each with zero as its Borel exceptional value. The present article proves related results in the case of rational coefficients. To this end, let \(D\) be a finite set of radii with arguments \(0 \leq \theta _1 < \cdots < \theta _m < \theta _{m+1} = \theta _{1}+2\pi\) and define \(\omega (D) = \max_{1 \leq j \leq m} \{\pi /(\theta _{j+1} - \theta _{j})\}\) and \(G(D,\epsilon) = \bigcup_{j=1}^m \{\text{arg} z - \theta_j < \epsilon \}\). For a meromorphic function \(f\), the zeros are attracted to \(D\), if for any \(\epsilon > 0\), \(n(r,G(D,\epsilon),1/f) = o(T(r,f))\) as \(r \to + \infty\). The main results are as follows: (a) If there exists a fundamental set of meromorphic solutions \(w_1,w_2,\ldots,w_n\) of (1) such that the zeros of each \(w_j\) are attracted to the rays in \(D\), then either \(\lambda = \max_{1 \leq j \leq n} \lambda {(w_j)} \leq \omega (D)\) or each \(w_j\) has zero as its Borel exceptional value of order \(\lambda\), i.e. for some \(\epsilon > 0\), \(n(r,1/f) < r^{\lambda - \epsilon}\). (b) If there exists a fundamental set of meromorphic solutions \(w_1,\ldots,w_n\) of (1) such that each \(w_j\) has its zeros attracted to the real axis, and at least one of these solutions with the maximal order \(\lambda\) does not have zero as its Borel exceptional value, then \(r_{j}(z) = a_{j} + O(1/z)\) for some \(a_…
reviewed by: Ilpo Laine
Mathematics Subject Classification ID: 34M05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 1004363
zbMATH Keywords: complex differential equations
zbMATH Keywords: asymptotic integration
zbMATH Keywords: Frank-Wittich conjecture
zbMATH Keywords: Borel exeptional value
zbMATH Keywords: fundamental set of meromorphic solutions
author: Jian-Hua Zheng
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04830501
OpenAlex ID: W2079625299
DOI: 10.2969/JMSJ/04830501
description: scientific article; zbMATH DE number 1004363