25
votes

Accepted

### Reference request: proof of Ramanujan's Cos/Cosh Identity

I expand my comment into an answer.
The key here is the Fourier series for the elliptic function $\operatorname {dn} (u, k) $ given as $$\operatorname {dn} (u, k) =\frac{\pi} {2K}\left(1+4\sum_{n=1}^{...

22
votes

Accepted

### Theta functions, re-expressed

Yes, it is true.
This generating function $\sum_n a_n q^n$
turns out to be the same as $(3F(q^3)-F(q))/2$:
they coincide through the $q^{100}$ term, which is more than enough
to prove equality ...

20
votes

Accepted

### A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$

Yes, it is always algebraic, because it is a modular function
evaluated at a CM (complex multiplication) point.
"$(q;q)_\infty$" is $q^{-1/24} \eta(\tau)$ where $q = e^{2\pi i \tau}$, so
"$(q;q)_\...

9
votes

Accepted

### How to work out this elliptic function?

This is a divergent series. But if one applies summation in the sense of Eisenstein,
$$\lim_{N\to\infty}\sum_{n=-N}^N\left(\lim_{M\to\infty}\sum_{m=-M}^M\right)$$
then the sum is doubly periodic. ...

9
votes

Accepted

### Constant term arising in general exact expression for the canonical height of a Mordell-Weil generator point on an elliptic curve $E$

You've decomposed the canonical height into a sum of local heights. If the curve has (split) multiplicative reduction at $p$, then $c_p=\operatorname{ord}_p(\Delta_E)$, the valuation of the minimal ...

8
votes

Accepted

### Generalizing Klein's order 7 formula $y\left(y^2+7\Big(\tfrac{1-\sqrt{-7}}{2}\Big)y+7\Big(\tfrac{1+\sqrt{-7}}{2}\Big)^3\right)^3 = j$ to order 13?

The smallest $m$ for which $\mathrm{PGL}(2,13)$ acts faithfully and transitively on a set of $13m$ elements is $m=6$, with point stabiliser a dihedral group of order $28$. I think you may be seeing a ...

8
votes

Accepted

### Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $

The question is whether this inequality on the Dirichlet eta function holds:
$$\Re\eta(\alpha+i\beta)\leq 2\eta(\alpha).$$
It does not hold.
Here is a plot of $\Delta(\alpha,\beta)=2\eta(\alpha)-\Re\...

6
votes

Accepted

### Can the following sum be counted or expressed in terms of special functions?

Probably the answer is negative. Your series is a restriction of the analytic function in two complex variables:
$$F(\zeta,q)=\sum_{n=0}^\infty\frac{q^{n^2}}{n!}\zeta^n,\quad |q|\leq1,$$
obtained by ...

6
votes

Accepted

### Finding order of vanishing for Jacobi Theta function

The convention when working with modular forms $f(\tau)$ of weight $k$ is for the order of vanishing at a cusp $a/c$ to mean the order of vanishing of $(c\tau+d)^{-k} f\left(\frac{a\tau+b}{c\tau+d}\...

5
votes

### What do theta functions have to do with quadratic reciprocity?

I think you're looking for the work of Tomio Kubota.
The square of the theta function is a modular form. For a while, and still today, the theta function itself is sometimes considered a modular form ...

5
votes

### Alternating power series $\sum_{k\geq 0}(-1)^k z^{(2k+1)^2}$

We can indeed prove, as already suspected by მამუკა ჯიბლაძე, that
$g(z)=\sum (-1)^k z^{(2k+1)^2}$ cannot be holomorphically continued past $z=1$. In fact, every point on $|z|=1$ is singular. This is a ...

5
votes

Accepted

### Jacobi forms and Kato's modular units

The triple product in the Jacobi theta function $\vartheta(\tau,z)$ can be rewritten
\begin{equation*}
\prod_{n \geq 1} (1-q^n) \prod_{n \geq 0} (1-q^n e^{2\pi i(z+\frac{\tau}{2}+\frac12)}) \prod_{n \...

5
votes

Accepted

### Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions

To expand on the comments. All four of the functions in the question are somehow versions of the classical theta function \begin{align*} \vartheta(\tau,z) &= \sum_{n \in \mathbb{Z}} (-1)^n q^{\...

5
votes

Accepted

### $q$-analog of an integral from quantum field theory?

The function
$$
f_q(x,y,z)=\sum_{cyc}e^z\frac{\theta_q\left(e^{\frac{2 \pi i}{3}+x-z}\right) \theta_q\left(e^{\frac{2 \pi i}{3}+y-z}\right)}{\theta_q\left(e^{x-z}\right) \theta_q\left(e^{y-z}\right)}...

5
votes

Accepted

### Approximation for a series involving the derivative of a Jacobi theta function

Write your $\Lambda$ as
$$\Lambda(t)=\frac{1}{2}\sum_{-\infty}^\infty f(n),$$
where
$$f(x)=\pi(x+1/2)\sin\pi(x+1/2)\exp\left(-\pi^2(x+1/2)^2t/2\right)=y\sin y\,e^{-ty^2/2},$$
where $y=\pi(x+1/2),$ and ...

5
votes

### Siegel modular forms in Mathematica

You might want to check out the documentation on Siegel Modular Forms by Yuen, Poor, Shurman, and King. It provides a variety of Mathematica and Maple notebooks.

5
votes

Accepted

### Is the left derivative of this theta function zero at $1$?

Writing $x=e^{-a}$ for $a>0$ and then using the Poisson summation formula with $s(u)=e^{i\pi u} u^2 e^{-au^2}$ (in the notation of the Wikipedia article), we rewrite the sum in question as
$$G(a):=\...

4
votes

Accepted

### How does $\prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5) $ hold?

First note that $\prod_{\xi} (1 - \xi^n q^n)$ is equal to $(1- q^n)^5$ if $5 | n$ and to $1 - q^{5n}$ otherwise. Moreover
$$
\varphi(q) = \prod_{n \geq 1} (1 - q^n)^{e_n}
$$
where $e_n = 1,-2,3,$ or $-...

4
votes

Accepted

### Entire function with almost periodic boundary condition?

The answer seems to be negative. Suppose that an entire function $f$
satisfies $f(z+v_i)=e^{A_iz}f(z)$, where $v_1$ and $v_2$ generate a lattice. Let $\Pi$ be the fundamental parallelogram of this ...

4
votes

Accepted

### On the Klein quartic and the similar $a^2b+b^2c+c^2a$?

The questions are:
Is there anything else special about $a^2b+b^2c+c^2a = 0$?
In what other contexts does it appear?
For question 1, Given the $q$-series of degree $9$,
$$ \frac ac=a^3-b^3,\quad\...

4
votes

Accepted

### 2D lattice sum with numerator

The $n=4$ sum is accessible as follows. Expand $(j+k)^4$; by symmetry
the odd terms $4j^3k$ and $4jk^3$ cancel out so we need only sum
$(j^4 + 6j^2k^2 + k^4) / (j^2+k^2)^4$. You say you already know
...

3
votes

Accepted

### Semidirect product of metaplectic group and Heisenberg group

I presume you are looking for a faithful action of $Mp_{2n}$ on something related to the Heisenberg group $H_{2n+1}$. This is well-known as Weil Representation.
In the modern language, consider $Mp_{...

3
votes

### " Laurent expansion" of quasi-periodic complex complex function

Bloch's theorem says that $f(z)=e^{iθz/a}u(z)$, with $u(z+a)=u(z)$. Then $u(z)=\sum_{n=-\infty}^\infty c_n \xi^n$ has a series expansion in powers of $\xi=e^{2\pi iz/a}$, which can be seen as a ...

3
votes

Accepted

### "Sparse" Theta Series

Won't the following argument show that the difference between successive exponents can never be bounded away from zero no matter how clever you try to be in selecting $(a,b)$?
The idea is to consider ...

3
votes

Accepted

### "One half of a theta-function" - is there something in the literature about it?

Here is one (but likely not the simplest) way to evaluate $F(t,q)+F(t^{-1},q)$. Equation (2.19) of
Milne, Stephen C., Infinite families of exact sums of squares formulas, Jacobi elliptic functions, ...

3
votes

### Obtain a series expansion of $a^2(q)a^2(q^4)$

The same group of authors work out the answer to your question in theorem 2 of this paper, so you can simply cite their result.
As mentioned in the comments, the answer gets more complicated because ...

3
votes

Accepted

### On $x^k+y^k=1$ and the Dixonian elliptic functions

Define the generalized trigonometic functions (discussion of these functions is given in this answer)
$$
z=\int_0^{\sin_{pr}z}\frac{dt}{\sqrt[p]{1-t^r}},\qquad \cos_{pr}z=\sqrt[r]{1-(\sin_{pr}z)^r},\...

3
votes

### Conformal blocks vector bundles on $\overline{M}_{g}$ in terms of generalized theta functions?

Under a certain natural compatibility assumption, there are counterexamples due to Belkale-Gibney-Kazanova.
The assumption essentially says that at a point $[\mathrm{C}]$ of the moduli space $\...

3
votes

Accepted

### Lower bound for the number of representations of integers as sum of squares

For $k=4$, your statement would be that $r_4(n) \gg n^{1-\epsilon}$. This is false. Jacobi's four-square theorem can be stated as that $r_4(n)/8$ is the sum of the divisors of $n$ that are not ...

3
votes

### Infinite product of $1-q^{n^2}$

Write $\theta(q)=\sum_{n \in \mathbb Z} q^{n^2}$. Taking logarithms, we get that the logarithm of your function is
$\sum_n \ln(1-q^{n^2}) = -\sum_n \sum_m \frac 1 m q^{n^2m} = -\sum_m \frac 1 m \frac {...

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