A note on the complex oscillation theory of non-homogeneous linear differential equations (Q757686)

It is proved the following theorem: Let \(a_ 0(z),...,a_{k-1}(z)\), \(P_ 0(z)\), \(P_ 1(z)\not\equiv 0\) be polynomials such that deg \(a_ 0=n\), deg (P\({}_ 0)=\beta <(n+k)/k\) and \(\deg (a_{k-j})\leq (\beta - 1)j\) for \(j=1,...,k-1\). Consider \[ (1)\quad f^{(k)}+a_{k- 1}(z)f^{(k-1)}+...+a_ 0(z)f=P_ 1(z)e^{P_ 0(z)}. \] a) If deg \(P_ 1<n\) then all solutions of (1) satisfy \[ (2)\quad \lambda (f)={\bar \lambda}(f)=\sigma (f)=(n+k)/k. \] b) If deg \(P_ 1\geq n\) then, apart from one possible exception, all solutions satisfy (2). The possible exceptional solution is of the form \(f=Q \exp (P_ 0)\) where Q is a polynomial of degree deg (P\({}_ 1)-n\), \(\sigma\) (f) is the order of the solution, \(\lambda\) (f) is the exponent of convergence.