Faà di Bruno’s Formula and Modular Forms

We show that Faà di Bruno’s formula can play important roles in modular forms theory and in the study of differential operators of the form a(x)ddxn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \displaystyle \left( a(x)\frac{d}{dx} \right) ^n$$\end{document}. We also emphasize the importance of the fundamental forms yk=Δ-k12,Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle y_k= \Delta ^{-\frac{k}{12}}, \Delta $$\end{document} is the discriminant function, making a link between some aspects of differential Galois theory and modular forms.