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Results tagged with modular-forms
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user 1703
Questions about modular forms and related areas
13
votes
1
answer
623
views
What relations exist among (quasi-) modular forms of different levels?
Given a (quasi-) modular form $f(\tau)$ for some congruence subgroup (say) $\Gamma(k)$, we know that $f(N\tau)$ is a (quasi-) modular form for $\Gamma(N k)$. Is there anything known about when we can …
- 6,242
2
votes
Expression for the derivative of Eisenstein series $G_2$
It's not too bad to check these out by hand, at least for a few low-degree examples such as $E_2$.
To be fair, we should be clear about normalizations before we begin: I will use the normalizations
…
- 6,242
6
votes
1
answer
276
views
Does the following operation on modular forms yield something modular?
Let $f(z) = \sum_{n=0}^\infty a_nq^n$ be the fourier expansion of a (quasi-)modular form (with $q = e^{2\pi i z}$). Consider the following related functions:
$$f_{m,k}(z) = \sum_{n=0}^\infty a_{mn + …
- 6,242
3
votes
1
answer
296
views
Is there a nice way to write the generating function obtained by taking the quadratic coeffi...
Suppose that you have a generating function
$$
f(q) = \sum_{k=0}^\infty a_k q^k
$$
It's not too hard to obtain the generating function
$$
f_{n,m}(q) = \sum_{k=0}^\infty a_{nk + m}q^k
$$
by taking a cr …
- 6,242
9
votes
1
answer
698
views
How do the rings of level $N$ quasi-modular forms related to the rings of modular forms?
It is well known that the graded algebra $\mathcal{M}(1)$ of Modular forms for $\Gamma = PSL_2(\mathbb{Z})$ is the polynomial algebra
$$
\mathcal{M}(1) = \mathbb{C}[E_4, E_6]
$$
where $E_4$ and $E_6$ …
- 6,242
4
votes
0
answers
105
views
Is there a nice way to invert this expression?
Let us first define the Euler polynomials to be the polynomials $P_n(q)$ that satisfy
$$
\frac{qP_n(q)}{(1 - q)^{n+1}} = \Big(q\frac{d}{dq}\Big)^n\frac{q}{1 - q}.
$$
For example, $P_0(q) = P_1(q) = 1 …
- 6,242
2
votes
Accepted
What literature is known about MacMahon's generalized sum-of-divisors function?
For anyone who comes by this later, it turns out that the following relationship is true for the functions $A_k(q)$:
$$
A_k(q) = \frac{1}{(2k+1)2k}\Big(\big(6A_1(q) + k(k-1)\big)A_{k-1}(q) - 2q\frac{d …
- 6,242
4
votes
2
answers
791
views
What literature is known about MacMahon's generalized sum-of-divisors function?
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 $a_{n,k}$ fo …
- 6,242
16
votes
3
answers
2k
views
How do modular forms of half-integral weight relate to the (quasi-modular) Eisenstein series?
The Eisenstein series
$$
G_{2k} = \sum_{(m,n) \neq (0,0)} \frac{1}{(m + n\tau)^{2k}}
$$
are modular forms (if $k>1$) of weight $2k$ and quasi-modular if $k=1$. It is clear that given modular forms $f, …
- 6,242