# How to find the solution of the equation $b=1+P(ba)$?

We know the solution(commutative case of Spitzer's Identity) of the equation $$b=1+\text{P}(ba)$$ when the operator $$\text{P}$$ satisfies Rota-Baxter eqution $$\text{P}(x)\text{P}(y)=\text{P}(x\text{P}(y))+\text{P}(\text{P}(x)y)+u\text{P}(xy)$$.(see Theorem 1.3.12 in the book "An introduction to Rota-Baxter Algebra" of Li Guo)

Is there anyway to find the solution of the equation $$b=1+\text{P}(ba)$$ when the operator $$\text{P}$$ satisfies $$\text{P}(x)\text{P}(y)=\text{P}(x\text{P}(y))+\text{P}(\text{P}(x)y)+uxy$$?

• Why do you put $P$ in text mode? it's not standard – YCor Mar 15 at 16:14