MEROMORPHIC SOLUTIONS TO NON-LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS

We consider the non-linear differential-difference equation c(z)w(z + 1) + a(z)w'(z)/w(z) = R(z, w(z)), where R(z, w(z)) is rational in w(z) with rational coefficients, a(z) and c(z) are non-zero rational functions. We give necessary conditions on the degree of R(z, w) for the above equation to admit a transcendental meromorphic solution of hyper-order rho 2(w) < 1. We also consider the admissible rational solutions of the above equation.