Differential equations satisfied by quasi modular forms?

Question

It is known that modular forms are solutions of differential equations. More precisely, let me cite the statement from the following question.

Differential Equations Satisfied by Modular Forms

Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{R})$ commensurable with $SL_{2}(\mathbb{Z})$. For $f \in M_{k}(\Gamma)$ (the space of weight $k$ modular forms) and $t \in M_{0}(\Gamma)$ (the space of meromorphic weight 0 modular forms), if $f = \sum_{n \geq 0}b_{n}t^{n}$ near $t = 0$, then there is a linear order $k + 1$ differential equation satisfied by $g(x) = \sum_{n \geq 0} b_{n}x^{n}$, of the form $$P_{k + 1}(x)\frac{d^{k + 1}g}{dx^{k + 1}} + P_{k}(x)\frac{d^{k}g}{dx^{k}} + \cdots + P_{0}(x)g = 0 \tag{1}$$

I seem to remember this is a theorem of Zagier, but I cannot give a precise reference. Anyway, my question is:

Are there a similar result for quasi-modular forms? I.e., if $f$ is only a quasi-modular form, is the above statement valid? If not, could anyone explain why? or give an example?

Answer 1

Let $\,f_0(x)\,$ be defined by $\,f_0(1/j(q)) = E_2(q),\,$ and $\, f_{n+1}(x) := x\frac{d}{dx}f_n(x).\,$ Then $$ 0 = 24x f_0(x) + (-816x+663552x^2)f_1(x) + (-1728x+2985984x^2)f_2(x) + (1-3456x+2985984x^2)f_3(x). \tag{1}$$ Check $\, f_0(x) = 1 - 24x -17928x^2 + O(x^3),\,\, f_1(x) = -24x -35856x^2 + O(x^3).\,$

An alternative form where $\, y := 1 - 1728x, \,\, f_{n+1}(x) := \frac{d}{dx}f_n(x)\,$ is

$$ 0 = 24f_0(x) + (1 -6000x +6635520x^2)f_1(x) + 3xy(1-2304x)f_2(x) + (xy)^2f_3(x). \tag{2}$$