I am looking at a proof that takes a Taylor series expansion around a (1 - $\alpha$) critical value $\mathscr{F}^\alpha$ for F-distribution $F_{p,K}$ around the corresponding Chi-square critical value $\chi^\alpha_p$. I am just confused on how to make sense on what I see below. I normally use Taylor series expansions for much nicer looking functions.
$$\mathscr{F}^\alpha \approx \frac{\chi^\alpha_p}{p} + \frac{G''_p(\chi^\alpha_p) (\chi^\alpha_p)^2}{pG'_p(\chi_p^\alpha)}\frac{1}{K} + o(b) $$
Where $G_p(z)$ is a Chi-square cdf function with degrees of freedom $p$ evaluated at $z$. I am a little lost at how we get each term.
I believe that by Slutsky's Theorem we can approximate F-distributions and F-critical values using Chi-Square, but this is just a bit beyond me at the moment.
UPDTAE: I am now convinced that this is actually using a Cornish-Fisher Type Expansion. Help is still appreciated. Thanks!