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A note on complex oscillation theory. (English) Zbl 0851.34004

In [Complex Variables, Theory Appl. 23, 101-121 (1993; Zbl 0801.34006)], the author 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 is 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 author’s idea to determine approximate square roots of a rational function is extended to determine approximate \(k\)th roots. Then \(k^2\) linear differential equations are 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 \(\lambda(f)< \infty\).
Reviewer: I.Laine (Joensuu)

MSC:

34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34A30 Linear ordinary differential equations and systems

Citations:

Zbl 0801.34006
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References:

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