Meromorphic solutions to non-linear differential-difference equations

Abstract

We consider the non-linear differential-difference equation $$ c(z)w(z+1)+a(z)\frac{w'(z)}{w(z)}=R(z,w(z)), $$ 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.

Keywords

• Meromorphic solutions
• differential-difference equations
• hyper-order