# Questions tagged [quasimodular-forms]

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### theta function with a low bound in the sum

I encounter a sum similar to the Jacobi Theta function except there is a lower bound $-J$ with $J\geq 0$: \begin{equation} f(q,x)=\sum_{n=-J}^\infty q^{n^2}x^n. \end{equation} My question is whether ...
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### Complex Multiplication and algebraic integers

Let $q=e^{2\pi i\tau}$ and $$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$ and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the ...
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### Power series which are $p$-adic modular forms for all $p$

Suppose that, for some integer $k$, a series $f(q) \in \mathbb Q \otimes \mathbb Z[[q]]$ has the property that for every prime $p$, $f(q)$ is the $q$-expansion of a $p$-adic modular form of weight $k$ ...
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### Algebraic independence of $E_2$, $E_4$ and $E_6$

In "M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, Prog. Math. 129, 165-172 (1995)" there is a proposition stating essentially that $E_2$, $E_4$ and $E_6$ are ...
228 views

### $f,f',…,f^{(j)}$ is $\mathbb C$-linearly independent if $f$ is a modular form

Conjecture: Let be $f$ a modular form of weight $k$ and $j$ a strictly positive integer, then the set $f,f',...,f^{(j)}$ is $\mathbb C$-linearly independent in $A$. Is that conjecture true or false? ...
565 views

### How do the rings of level $N$ quasi-modular forms related to the rings of modular forms?

It is well known that the graded algebra $\mathcal{M}(1)$ of Modular forms for $\Gamma = PSL_2(\mathbb{Z})$ is the polynomial algebra $$\mathcal{M}(1) = \mathbb{C}[E_4, E_6]$$ where $E_4$ and $E_6$ ...
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### Binomial coefficients and derivatives of modular forms

Let $E_2$, $E_4$, and $E_6$ denote the standard Eisenstein series. The usual variables $q=e^{2\pi i\tau}$ allow us to regard the $E_n$'s as functions on either the upper half plane or the unit disk ...
MacMahon in the paper Divisors of Numbers and their Continuations in the Theory of Partitions defines several generalized notions of the sum-of-divisors function; for example, if we write $a_{n,k}$ ...