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For the first-order differential-difference equations of the formA(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),whereA(z),B(z),C(z)A\left(z),B\left(z),C\left(z), andF(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 whendegB(z)&lt;deg{A(z)+C(z)}+1{\rm{\deg }}B\left(z)\lt {\rm{\deg }}\left\{A\left(z)+C\left(z)\right\}+1and all transcendental solutions are of order at least 1. For the finite-order transcendental solutionf(z)f\left(z), the relationship betweenρ(f)\rho (f)andmax{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.

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