Why any quasi modular form corresponds to some almost holomorphic form?

Question

I am trying to understand this article.

http://arxiv.org/pdf/math/0603268v1.pdf

In particular Theorem 1. In particular it says that if

$$ (c z + d)^{-k} f_0(\frac{a z + b} {c z + d}) = \sum_{j = 0}^{p} f_j(z) (\frac{c} {c z + d})^j$$

Then $$ (c z + d)^{-k + 2 l} f_l( \frac{a z + b} {c z + d}) = \sum_{j \geq l} \binom{j} {l} f_j(z) (\frac{c} {c z + d})^{j - l}$$

First equation is special case $l=0$.

Question: How to prove this?

Comment: Author give link to this article, but I did not find a proof. http://people.mpim-bonn.mpg.de/zagier/files/progmath/129/165/fulltext.pdf

Answer 1

By definition, quasi-modular forms are constant terms of almost holomorphic modular forms w.r.t. $y={\rm Im}(\tau)$ expansion. Therefore, the map $\sum_{i=0}^n f_i(\tau) y^{-i} \rightarrow f_0(\tau)$ is surjective. Proof of injectivity can be found in this paper

http://arxiv.org/pdf/alg-geom/9712009.pdf

(see Proposition 3.4)