On the growth of meromorphic solutions of some higher order linear differential equations

Let k, m , n be integers such that k >= 1, n >= 2 and 1 <= m <= n. In this article we study the order rho(f) and the hyperorder rho(2)(f) of nonzero meromorphic solutions f of the differential equation Sigma(n)(j=1, j not equal m) A(j)(z)f((j))(z) + A(m)(z)e(pm(z)) f((m))(z) + (A(0)(z)e(p(z) )+ B-0(z)e(q(z))) f(z) = 0, where B-0(z), A(0)(z), . . . , A(n)(z) are meromorphic functions such that A(0)A(m)A(n)B(0) not equivalent to 0, max{rho(B-0), rho(A(0)), . . . , rho(A(n))} < k, and p(z), q(z), p(m)(z) are polynomials of degree k. Under some conditions, we show that rho(f) = +infinity and rho(2)(f) = k . This is an extension of some recent results by Peng, Chen, Xu, and Zhang devoted to linear differential equations of the second order.