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

Abstract

In this paper, we investigate the order and the hyper order of entire solutions of the higher order linear differential equation f(k) +Ak−1 (z) e Pk−1(z) f(k−1)+...+A1 (z) e P1(z) f

+A0 (z) e P0(z) f =0 (k ≥ 2) , where Pj (z) (j = 0, ..., k − 1) are nonconstant polynomials such that deg Pj = n (j = 0, ..., k − 1) and Aj (z) (≡ 0) (j = 0, ..., k − 1) are entire functions with ρ (Aj ) < n (j = 0, ..., k − 1). Under some conditions, we prove that every solution f (z) ≡ 0 of the above equation is of infinite order and ρ2 (f) = n.