On the uniqueness problems of entire functions and their linear differential polynomials

The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results on this topic have been obtained. In this paper, we study a transcendental entire function f(z) that shares a non-zero polynomial a(Z) with f'(z), together with its linear differential polynomials of the form: L[f] = a(2) (Z) f" (Z) + a(3) (Z) f"' (Z) + ... + a(m) (z)f ((m)) (z) (a(m) (z) not equivalent to 0), where the coefficients a(k)(Z) (k = 2, 3,..., m) are rational functions.