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 Br¨uck’s conjecture, that is, if ∆f (z) = f (z+η)-f (z) and f (z) share one value a ( 6=α) CM, whereη∈C is a constant such that f (z+η) 6≡f (z), then∆f (z)-af (z)-a = a a-α.
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