Entire functions that share a polynomial with their derivatives ✩

Let f be a nonconstant entire function, k and q be positive integers satisfying k >q, and let Q be a polynomial of degree q. This paper studies the uniqueness problem on entire functions that share a polynomial with their derivatives and proves that if the polynomial Q is shared by f and f CM, and if f (k)(z) − Q(z) = 0 whenever f (z) − Q(z) = 0, then f ≡ f . We give two examples to show that the hypothesis k >q is necessary. © 2005 Elsevier Inc. All rights reserved