Differential polynomials generated by second order linear differential equations. (English) Zbl 1165.34430

Belaïdi, B.; El Farissi, Abdallah

J. Appl. Anal. 14, No. 2, 259-271 (2008).

Summary: We study fixed points of solutions of the differential equation
\[ f^{{\prime \prime }}+A_{1}( z) f^{{\prime }}+A_{0}(z) f=0, \] where \(A_{j}(z) ( \not\equiv 0)\), \(j=0, 1\), are transcendental meromorphic functions with finite order. Instead of looking at the zeros of \(f( z)-z\), we proceed to a slight generalization by considering zeros of \(g(z)-\varphi ( z)\), where \(g\) is a differential polynomial in \(f\) with polynomial coefficients, \(\varphi\) is a small meromorphic function relative to \(f\), while the solution \(f\) is of infinite order.

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

Keywords:

linear differential equations; meromorphic solutions; hyper order; exponent of convergence of the sequence of distinct zeros; hyper exponent of convergence of the sequence of distinct zeros

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