# Generalization of Ramanujan's formulas for derivatives of Eisenstein series

Let $$E_2, E_4,$$ and $$E_6$$ be the first three Eisenstein series. There are well-known formulas due to Ramanujan for the derivatives of these quantities:

\begin{align*} DE_2 &= \frac{1}{12}(E_2^2-E_4) \\ DE_4 &= \frac{1}{3}(E_2E_4-E_6) \\ DE_6 & = \frac{1}{2}(E_2E_6-E_4^2), \end{align*} where $$(Df)(\tau):=\frac{1}{2\pi i}f'(\tau)$$. A noticeable feature of these identities is that, for each $$k=2,4,6$$, there are constants $$a_k, b_k$$ such that $$DE_k - a_kE_2E_k = b_kE_{k+2}.\tag{*}$$ I was wondering if a formula of this type held for higher $$k$$. Now, I know that given a modular form $$f\in\mathcal{M}_k$$ with $$k\geq 4$$, the so-called Serre derivative $$\vartheta f = Df - \frac{k}{12}E_2f$$ has the property $$\vartheta f\in\mathcal{M}_{k+2}$$. If $$\mathcal{M}_{k+2}$$ happens to be one-dimensional, then a formula of the type (*) indeed holds with $$a_k=k/12$$. This means the minimal $$k$$ for which such a formula might fail is $$k=10$$ (so that $$\mathcal{M}_{k+2} = \mathcal{M}_{12}$$ is two-dimensional). In principle I know one can calculate a Fourier expansion for $$\vartheta E_{10}$$ and then write it in the basis of $$\mathcal{M}_{12}$$ by comparing coefficients, but I am having a hard time doing the calculation.

Even if (*) turns out to be false, I wonder if a similar-looking recurrence relation might still hold. Perhaps this is well-known result (I know very little about modular forms), but I couldn't find anything in my cursory glance of the literature.

So, I essentially have four questions:

1. Does (*) hold for $$k=10$$?
2. What about for higher $$k$$?
3. Even if (*) is false, is there still some generalized recurrence formula on the derivatives?
4. And if the answer to the previous question is "yes," where might I find the details?
• For $k \le 20$ it suffices to check the first two coefficients (if $h$ of weight $< 24$ has a double zero at $i\infty$ then $\Delta^{-2} h$ is a weight $< 0$ modular form so it is $0$) Feb 11 at 11:42

I conjectured that $$D E_k - a E_{k+2} = \sum_m b_m E_m E_{k+2-m}$$ for some integers $$a$$ and $$b_m$$ depending on $$k$$ assuming a basis for $$\mathcal{M}_{k+2}$$ and then solved for those integers. My numerical results include: $$D E_8 = (E_2 E_8 - E_{10})2/3,$$ $$D E_{10} = (735 E_2 E_{10} - 44E_6 E_6 - 691 E_{12})/882,$$ $$D E_{12} = (E_2 E_{12} - E_{14}),$$ $$D E_{14} = (3773 E_2 E_{14} - 156E_6 E_{10} - 3617 E_{16})/3234,$$ $$D E_{16} = (39787 E_2 E_{16} + 4080 E_4 E_{14} - 43867 E_{18})4/119361,$$ $$D E_{18} = (5307907 E_2 E_{18} - 244188 E_6 E_{14} - 5063719 E_{20})3/10615814,$$ $$D E_{20} = (1890862519 E_2 E_{20} - 90714360 E_6 E_{16} - 1800148159 E_{22})5/5672587557.$$ Other similar expressions can be found but I have no proofs. The first case where more than two nonzero terms are needed in the finite summation is when $$k=22$$ and more than three nonzero terms are needed when $$k=34$$.
• This is a start! Are you just using that $E_mE_{k+2-m}$ form a basis for $\mathcal{M}_{k+2}$ to get the decomposition, or did you have a more explicit method for getting the coefficients? Feb 12 at 23:14