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

Abstract

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),A\left(z)f\left(z+1)+B\left(z)f^{\prime} \left(z)+C\left(z)f\left(z)=F\left(z), where A(z),B(z),C(z)A\left(z),B\left(z),C\left(z), and F(z)F\left(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 degB(z)<deg{A(z)+C(z)}+1{\rm{\deg }}B\left(z)\lt {\rm{\deg }}\left\{A\left(z)+C\left(z)\right\}+1 and all transcendental solutions are of order at least 1. For the finite-order transcendental solution f(z)f\left(z), the relationship between ρ(f)\rho (f) and max{λ(f),λ(1∕f)}\max \left\{\lambda (f),\lambda \left(1/f)\right\} is discussed. Some examples for sharpness of our results are provided.

Keywords

• differential-difference equation
• growth and zeros
• meromorphic solution
• 30d35
• 39a06