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On the hyper order of solutions of a class of higher order linear differential equations. (English) Zbl 1199.34468
Summary: We investigate the order and the hyper order of entire solutions of the higher order linear differential equation \[ f^{(k)}+A_{k-1}(z) e^{P_{k-1} (z)} f^{(k-1)} +\dots +A_1 (z) e^{P_1 (z)} f'+A_0 (z) e^{P_0 (z)} f=0\;(k\geq 2), \] where \(P_j (z)\;(j=0,\dots ,k-1)\) are nonconstant polynomials such that deg \(P_j =n\;(j=0,\dots ,k-1)\) and \(A_j (z)\;(\not\equiv 0)\;(j=0,\dots ,k-1)\) are entire functions with \(\rho(A_j )<n\;(j=0,\dots ,k-1)\). Under some conditions, we prove that every solution \(f(z)\not\equiv 0\) of the above equation is of infinite order and \(\rho _2 (f)=n\).

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
34M03 Linear ordinary differential equations and systems in the complex domain
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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