On the order and hyper-order of meromorphic solutions of higher order linear differential equations

Abstract

In this paper, we investigate the order of growth of solutions of the higher order linear differential equation

f^(k) + Σ^k-1_j=0 (h_je^P_j(z) + d_j) f^(j) = 0,

where P_j(z) (j = 0,1,…,k-1) are nonconstant polynomials such that deg P_j = n ≥ 1 and h_j(z), d_j(z) (j = 0,1,…,k-1) with h₀ ≢ 0 are meromorphic functions of finite order such that max {ρ (h_j),ρ(d_j): j = 0,1,…,k-1} < n. We prove that every meromorphic solution f ≢ 0 of the above equation is of infinite order. Then, we use the exponent of convergence of zeros or the exponent of convergence of poles of solutions to obtain an estimation of the hyper-order of solutions.