Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $.

We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$.

Do you have any idea how to obtain this Taylor series expansion formula ? $$F^{-1}_{Y+\lambda Z}(\alpha)=F^{-1}_{Y}(\alpha)+\lambda[\frac{dF^{-1}_{Y+\lambda Z}}{d\lambda}(\alpha)]_{\lambda = 0} + \frac{\lambda^2}{2!}[\frac{d^2 F^{-1}_{Y+\lambda Z}}{d\lambda^2}(\alpha)]_{\lambda = 0}+...+\frac{\lambda^n}{n!}[\frac{d^n F^{-1}_{Y+\lambda Z}}{d\lambda^n}(\alpha)]_{\lambda = 0}+...$$