On the hyper-order of solutions of a class of higher order linear differential equations

In this paper, we investigate the growth of solutions of the linear differential equation \begin{multline*} f^{(k)}+\left( A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}\left( z\right) \right) f^{(k-1)}+\cdots +\\\left( A_{1}(z)e^{P_{1}(z)}+B_{1}\left( z\right) \right) f^{\prime } +\left( A_{0}(z)e^{P_{0}(z)}+B_{0}\left( z\right) \right) f=0, \end{multline*} where $k\geq 2$\ is an integer, $P_{j}(z)$ $(j=0,1,\cdots ,k-1)$\ are nonconstant polynomials\ and $A_{j}(z)$ $\left( \not\equiv 0\right) ,$ $ B_{j}\left( z\right) $\ $\left( \not\equiv 0\right) $ $(j=0,1,\cdots ,k-1)$\ are meromorphic functions. Under some conditions, we determine the hyper-order of these solutions.