An entire function and its derivatives sharing a polynomial (Q874922)

In the paper the authors prove that if \(Q_{1}\), \(Q_{2}\) are polynomials of degree at least \(1\) and \(f\) is a non-constant solution of \(f' - Q_{1} = e^{P}(f - Q_{2})\) then the hyper-order of \(f\) is equal to the degree of \(P\), where \(P\) is a polynomial. As consequences of this result several uniqueness theorems for a non-constant entire function sharing certain polynomial with its derivative are deduced. These results are related to a conjecture of R. Brück.