In this paper, the authors investigate a delay differential equation of the form \(w({z + 1}) - w({z - 1}) + a(z){{w^{\prime}(z)} \over {w(z)}} = {{P({z,w})} \over {Q({z,w})}},\) where a(z) is a nonzero rational function, P(z, w) and Q(z, w) are prime polynomials in w with rational coefficients. They remove the restriction that the order of meromorphic solutions of the above difference equation is σ2(w) < 1, and obtain the growth of transcendental meromorphic solutions. The exact forms of all transcendental entire solutions are obtained when degw P = degw Q = 0, or degw P = 1 and degw Q = 0, respectively. If degw P ≥ 2 and degw Q = 0, or degw Q ≥ 1 and Q(z, 0) ≢ 0, they prove that the above equation has no transcendental entire solution. They show that the existence of transcendental entire solutions of the above equation depends on the degrees of P(z, w) and Q(z, w).
