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\begin{document}
\section{Quasi-modular forms}
Let $f_0, f_1, \dots, f_n$ be functions of $z$ on $\HH$. We consider
\[
f(z, X) = \sum_i \frac{f_i(z)}{i!(z-X)^i}.
\]
An element of the group $SL_2(\R)$ acts in weight $w$ in $z$ and weight $0$ in $X$:
\begin{multline*}
f(z, X)|_w \gamma = \sum_i \frac{f_i(\gamma z) (cz+d)^{i-w} (cX+d)^i}{i! (z-X)^i}\\
=\sum_{i, j} \frac{f_i(\gamma z) (cz+d)^{i-w} \binom{i}{j} (cX-cz)^{i-j} (cz+d)^j}{i! (z-X)^i}\\
=\sum_{i, j} \frac{(-1)^{i-j}}{(i-j)!} f_i(\gamma z) (cz+d)^{i+j-w} c^{i-j} \frac{1}{j!(z-X)^j} \\
=\sum_{i, j} (-1)^{i-j} f_i(z)|_{w-2i} \gamma \frac{1}{(i-j)!}\left(\frac{c}{cz+d}\right)^{i-j} \frac{1}{j!(z-X)^j}.
\end{multline*}
Therefore, expanding the result as a polynomial in $(z-X)^{-1}$ we get
\[
f(z, X)|_w \gamma = \sum_i \frac{g_i(z)}{i!(z-X)^i},
\]
where
\[
g_i(z) = \sum_j f_{i+j}(z)|_{w-2i-2j} \gamma \frac{(-1)^{j}}{j!} \left(\frac{c}{cz+d}\right)^j.
\]
Therefore we define the action of the group on tuples of functions as follows:
\[
(f_0, f_1, \dots, f_n)|_w \gamma = (g_0, g_1, \dots, g_n),
\]
where $g_i$ are given by the formula above. It is clear, that $g_i$ depends only on $f_i$, $f_{i+1}$, \dots, $f_n$,
so this action is an iterated exension of the usual action on functions.
\begin{defn}
A quasi-modular form of weight $w$ and depth $n$ for group $\Gamma$ is a holomorphic function $f$ on $\HH$, which has moderate growth at the cusps and such, that there exist functions $f_1$, \dots, $f_n$ such, that the tuple $(f, f_1, \dots, f_n)$ is invariant under the action of $\Gamma$ in weight $w$.
\end{defn}
\begin{rem}
It implies, that the each function $f_i$ is again a quasi-modular form of weight $w-2i$ and depth $n-i$.
\end{rem}
Looking at the definition of the action we can express $f_i(z)|_{w-2i}\gamma$ in terms of $g_i(z)$:
\[
f_i(z)|_{w-2i} \gamma = \sum_j g_{i+j}(z) \frac{1}{j!} \left(\frac{c}{cz+d}\right)^j.
\]
We prove this as follows:
\begin{multline*}
\sum_j g_{i+j}(z) \frac{1}{j!} \left(\frac{c}{cz+d}\right)^j \\
= \sum_{j, l} \frac{1}{j!} \left(\frac{c}{cz+d}\right)^j f_{i+j+l}|_{w-2i-2j-2l}\gamma \frac{(-1)^l}{l!} \left(\frac{c}{cz+d}\right)^l \\
= \sum_{m} f_{i+m}|_{w-2i-2m}\gamma \left(\frac{c}{cz+d}\right)^m \sum_{l=0}^m \frac{(-1)^l}{l!(m-l)!} = f_i|_{w-2i} \gamma.
\end{multline*}
So, one can prove, that
\begin{prop}
The function $f$ is quasi-modular of weight $w$, depth $n$ for a group $\Gamma$ if and only if is holomorphic, of moderate growth at the cusps and
\[
f(z)|_w\gamma = \sum_{i=0}^n f_i(z) \frac{1}{i!} \left(\frac{c}{cz+d}\right)^i
\]
for some functions $f_i$ and for all $\gamma\in\Gamma$. Moreover, if this is true, than the functions $f_i$ are the functions, which are needed in the definition and $f_0=f$.
\end{prop}
Let $f_0$, $f_1$, \dots, $f_n$ be again functions of $z$ and
\[
f(z, X) = \sum_i \frac{f_i(z)}{i!(z-X)^i}.
\]
We put 
\[
\delta_w f(z, X) = \frac{\partial f(z, X)}{\partial z} + \frac{w f(z, X)}{z-X},
\]
this is a derivation (taking the weight into account), and it commutes with the action of the group.
The coefficients of $\delta_w f(z, X)$ are given by
\[
(\delta_w f(z, X))_i = \frac{\partial f_i(z)}{\partial z} + i(w+1-i) f_{i-1}(z),
\]
we see, in particular, that the first coefficient is simply the derivative of the original first coefficient, and the depth is increased by $1$.
Another differential operator is 
\[
\delta_w^- f(z, X) = (z-X)^2 \frac{\partial f(z, X)}{\partial X},
\]
it also is a derivation and commutes with the action of the group. The coefficients are
\[
(\delta_w^- f(z, X))_i = f_{i+1},
\]
so this operator 'forgets' the first coefficient and decreases depth by $1$. Their commutator is
\[
(\delta_{w+2}^-\delta_w - \delta_{w-2}\delta_w^-) f(z, X) = w f(z, X).
\]
On the level of quasi-modular forms the first operator is simply the derivative with respect to $z$, and the second maps $f$ to $f_1$ (see the definition).
Since there is a group action on tuples $(f_0, f_1, \dots, f_n)$ we have an action of the Lie algebra. For a function of $z$ and $X$ one can compute:
\[
L_wf = \frac{\partial}{\partial t}|_{t=0} (f|_w\begin{pmatrix}1&t\\0&1\end{pmatrix}) = \frac{\partial f}{\partial z} + \frac{\partial f}{\partial X},
\]
\[
M_wf = \frac{\partial}{\partial t}|_{t=0} (f|_w\begin{pmatrix}1&0\\t&1\end{pmatrix})
= -w z f - z^2 \frac{\partial f}{\partial z} - X^2 \frac{\partial f}{\partial X}.
\]
\[
(L_wM_w - M_w L_w)f = -wf - 2z\frac{\partial f}{\partial z} - 2X\frac{\partial f}{\partial X}.
\]

\end{document}@