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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?

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  • $\begingroup$ You mean like Ramanujan identities for Eisenstein series? $\endgroup$ – Somos Feb 18 '19 at 1:09
  • $\begingroup$ @Somos No. A Ramanujan identity involve all the three $E_2$, $E_4$ and $E_6$. I mean, e.g. for $E_2$, whether there is a linear third order ODE of the above form which has $E_2$ as a solution. $\endgroup$ – Wenzhe Feb 18 '19 at 3:09
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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

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

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