Hyper-order estimates for transcendental meromorphic solutions of linear complex differential equations

Abstract

In this paper, we give some results about of the growth of transcendental meromorphic solutions of linear differential equations%
\begin{equation*}
f^{\left( k\right) }+\sum_{j=0}^{k-1}\left( D_{j}\left( z\right)
+A_{j}\left( z\right) e^{P_{j}\left( z\right) }+B_{j}\left( z\right)
e^{Q_{j}\left( z\right) }\right) f^{\left( j\right) }=F,
\end{equation*}%
where $A_{j}\left( z\right) \left( \not\equiv 0\right) ,$ $B_{j}\left( z\right) \left( \not\equiv 0\right) $ and $D_{j}\left( j=0,1,...,k-1\right) $ are meromorphic functions with finite order,\textit{\ }$P_{j}\left( z\right)
=a_{j,n}z^{n}+\cdots +a_{j,0}$\textit{\ }and\textit{\ }$Q_{j}\left( z\right) =b_{j,n}z^{n}+\cdots +b_{j,0}$\textit{\ }$\left( j=0,...,k-1\right) $\ are nonconstant polynomials, $a_{j,0},...,a_{j,n},b_{j,0},...,b_{j,n}$ ($%
j=0,1,...,k-1$)\textit{\ }are complex numbers. Under some conditions, we prove that the hyper-order of every transcendental meromorphic solution of the above equation and of its corresponding homogeneous equation coincides with $n$.