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

In this paper, we investigate the order and the hyper order of entire solutions of the higher order linear differential equation f((k)) + A(k-1) (z) e(Pk-1(z)) f((k-1)) +...+ A(1) (z) e(P1(z)) f' + A(0) (z) e(P0(z)) f = 0 (k >= 2) where P-j (z) (j = 0,..., k - 1) are nonconstant polynomials such that deg P-j = n (j = 0,..., k - 1) and A(j) (z) (not equivalent to 0) (j = 0,..., k - 1) are entire functions with rho (A(j)) < n (j = 0,..., k - 1). Under some conditions, we prove that every solution f (z) not equivalent to 0 of the above equation is of infinite order and rho(2) (f) = n.