Let $P$ be an operator such that $P(kx)=kP(x)$, $k \in \mathbb{C}$, $x$ is a variable, $P(xy)=P(xP(y))+P(P(x)y)-P(x)P(y)$, $x, y$ are variables. All variables commute.
Let $P(t)=t$. Then $P(t^2)=2P(tP(t))-P(t)P(t)=2P(t^2)-t^2$. Hence $P(t^2)=t^2$.
Let $P(t)=t^2$. Then $P(t^2)=P(tP(t)) + P(P(t)t)-P(t)P(t)=2P(t^3)-t^4$. Is it possible to compute $P(t^2)$? Thank you very much.