Properties of meromorphic solutions of first-order differential-difference equations

For the first-order differential-difference equations of the form A(z)f (z + 1) + B(z)f '(z) + C(z)f (z) =F(z),where A(z), B(z), C(z), and F(z) are polynomials, the existence, growth, zeros, poles, and fixed points of their nonconstant meromorphic solutions are investigated. It is shown that all nonconstant meromorphic solutions are transcendental when deg B(z) < deg{A(z) + C(z)} + 1 and all transcendental solutions are of order at least 1. For the finite-order transcendental solution f(z), the relationship between rho(f) and max{lambda(f), lambda(1/f)} is discussed. Some examples for sharpness of our results are provided.