Linear differential equations with rational coefficients
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 \(a_j \in \mathbb{C}\) as \(|z|\to + \infty\) and either at least one \(a_j \neq 0\) or \(\lambda (w_j) = \frac{1}{2}\) for each \(j\). The proofs rely on the use of asymptotic integration and the indicator function.
fundamental set of meromorphic solutions
scientific article; zbMATH DE number 1004363