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 deg B ( 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.