Hello

I am trying to follow the derivation of the Generalized Cornish Fisher Expansion from this paper.

I dont see how the author goes from equation (21) to (22).

For one to be able to do this consider an arbitrarily differentiable function $G(v)$.

Make the change of variable $v = \Phi(u)$ which implies $u=\Phi^{-1}(v)$. Note that $\frac{du}{dv}=\frac{1}{\phi(u)}$. Let $D_x=\frac{d}{dx}$ the differential operator.

Now equation (21) would imply (22) if

$D_v^r[G(v)]= \left(D_u\frac{\text{du}}{\text{dv}}\right){}^r=D_u^r\left(\frac{\text{du}}{\text{dv}}\right)^r$

Is this true? and if so how?

It is clearly true for $r=1$ by the chain rule. But for $r=2,3,...$ I don't see how.

Going from (22) to (23) is also a mystery to me.