On the difference counterpart of br¨ck's conjecture

In this article, for a transcendental entire function f(z) of finite order which has a finite Borel exceptional value α, we utilize properties of complex difference equations to prove the difference counterpart of Brück's conjecture, that is, if Δ f(z)=f(z+η)-f(z) and f(z) share one value a(≠α) CM, where η ∈ ℂ is a constant such that f(z+η)≡ f(z), then Δf(z)−af(z)−a=aa−α.