Non-Linear Complex Differentialdifference Equations Admit Meromorphic Solutions

Abstract

We obtain necessary conditions for the non-linear complex differential-difference equations $$w(z + 1)w(z - 1) + a(z)\frac{{{w'}(z)}}{{w(z)}} = R(z,w(z))$$ to admit transcendental meromorphic solutions w(z) such that ρ2(w) < 1, where R(z,w(z)) is rational in w(z) with rational coefficients, a(z) is a rational function and ρ2(w) is the hyper-order of w(z). Our results can be seen as the product versions on an equation of another type investigated by Halburd and Korhonen [3]. We also provide an idea which implies that the case of degw(R(z,w)) = 4 in the original proof of [3, Theorem 1.1] can be organized in a short way.