Growth of meromorphic solutions of some linear differential equations

In this paper, we investigate the order and the hyper-order of meromorphic solutions of the linear differential equation f((k)) + Sigma(k-1)(j=1) (D-j + B(j)e(Pj(z)))f((j)) + (D-0 + A(1)e(Q1(z)) + A(2)e(Q2(z)))f = 0, where k >= 2 is an integer, Q(1)(z); Q(2)(z), P-j(z) (j = 1, ... , k - 1) are nonconstant polynomials and A(s) (z) (not equivalent to 0) (s = 1, 2), B-j (z) (not equivalent to 0) (j = 1, ... , k - 1), D-m(z) (m = 0, 1, ... , k - 1) are meromorphic functions. Under some conditions, we prove that every meromorphic solution f (not equivalent to 0) of the above equation is of infinite order and we give an estimate of its hyper-order. Furthermore, we obtain a result about the exponent of convergence and the hyper-exponent of convergence of a sequence of zeros and distinct zeros of f - phi, where phi (not equivalent to 0) is a meromorphic function and f (not equivalent to 0) is a meromorphic solution of the above equation.