GROWTH OF SOLUTIONS TO HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH ENTIRE COEFFICIENTS

In this article, we discuss the order and hyper-order of the linear differential equation f((k)) +Sigma(k-1)(j=1)(B(j)e(bjz) + D(j)e(djz))f((j)) + (A(1)e(a1z) + A(2)e(a2z))f = 0, where A(j)(z), B-j(z), D-j(z) are entire functions (not equivalent to 0) and a(1), a(2), d(j) are complex numbers (not equivalent to 0), and b(j) are real numbers. Under certain conditions, we prove that every solution f not equivalent to 0 of the above equation is of infinite order. Then, we obtain an estimate of the hyper-order. Finally, we give an estimate of the exponent of convergence for distinct zeros of the functions f((j)) - phi (j = 0, 1, 2), where phi is an entire function (not equivalent to 0) and of order sigma(phi) < 1, while the solution f of the differential equation is of infinite order. Our results extend the previous results due to Chen, Peng and Chen and others.