Entire functions that share fixed points with finite weights (Q657669)

Let \(f\) be a transcendental entire function and let \(P\) be a polynomial. Using Nevanlinna theory, the author considers fixed points of differential polynomials of the form \((f^nP(f))^{(k)}\). Write \(P(Z)=a_mZ^m+a_{m-1}Z^{m-1}+\dots+a_1Z+a_0\), where \(0\neq a_0, a_1, \dots, a_{m-1}, a_m \neq 0\) are complex constants. It is shown that if \(n\geq k+2\) then \((f^nP(f))^{(k)}\) has infinitely many fixed points. The author also considers the uniqueness problem for two entire functions \(f\) and \(g\) with some weighted sharing conditions on \((f^nP(f))^{(k)}\) and \((g^nP(g))^{(k)}\).