I am reading Avrachenkov's book "Analytic perturbation theory and its applications". One of the problems contained in the book is as follows:
If we substitute the series expansions $$A(z)=A_0+zA_1+z^2A_2+\dots$$ and $$A^{-1}(z)=\frac{1}{z^s}(X_0+zX_1+z^2X_2+\dots)$$ into the equation $A(z)A^{-1}(z)=I,$ we obtain the set of equations $$\sum_{i=0}^kA_iX_{k-i}=\delta_{ks}I,\quad k=0,1,\dots\tag{1}$$ where $\,\delta_{ks}\,$ is the Kronecker delta. And if we substitute the above series into the equation $A^{-1}(z)A(z)=I,$ we obtain the set of equations $$\sum_{i=0}^k X_{k-i}A_i=\delta_{ks}I,\quad k=0,1,\dots\tag{2}$$ How can one prove that the set of equations $(1), (2)$ are equivalent?
I would be grateful for any hints or solutions for this problem, because all my attempts are bad.