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Zeros of meromorphic solutions of higher order linear differential equations. (English) Zbl 0815.34003

The author studies the zeros of meromorphic solutions of the differential equation \(f^{(k)} + A_{k-1} f^{(k-1)} + \cdots + A_ 0f = F\) where \(F\) and the \(A_ j\) are meromorphic functions that do not vanish identically. A typical result is Theorem 1: If \(F\) and \(A_ 1, \dots, A_{k-1}\) have finite order and \(A_ 0\) has infinite lower order but the zeros of \(A_ 0\) have finite exponent of convergence, then there exists at most one solution for which the exponent of convergence of the distinct zeros is finite. The proofs are based on Nevanlinna’s theory on the distribution of values.

MSC:

34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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