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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 line 21 equation (21) to 22. (22).

For one to be able to do this consider an arbitrarily differential 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 wouldn't seem like it from is clearly true for $r=1$ by the Faa di Bruno formulachain rule. But for $r=2,3,...$ I don't see how.

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

2 corrected spelling in the title

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# Some help with differntial operators

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 line 21 to 22.

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

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

Now (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 wouldn't seem like it from the Faa di Bruno formula.

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