26
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 $$\operatorname {dn} (u, k) =\frac{\pi} {2K}\left(1+4\sum_{n=1}^{...
- 2,249
22
votes
Accepted
Theta functions, re-expressed
Yes, it is true.
This generating function $\sum_n a_n q^n$
turns out to be the same as $(3F(q^3)-F(q))/2$:
they coincide through the $q^{100}$ term, which is more than enough
to prove equality ...
- 79.9k
20
votes
Accepted
A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$
Yes, it is always algebraic, because it is a modular function
evaluated at a CM (complex multiplication) point.
"$(q;q)_\infty$" is $q^{-1/24} \eta(\tau)$ where $q = e^{2\pi i \tau}$, so
"$(q;q)_\...
- 79.9k
16
votes
Why does this theta function value yield such a good Riemann sum approximation?
This is because the theta function has a functional equation.
Under the usual definition $\theta(t) = \sum_n e^{-\pi t n^2}$, the functional equation is $$\theta(t) = \frac 1{\sqrt t} \theta\left(\...
- 3,432
14
votes
Accepted
Why does this theta function value yield such a good Riemann sum approximation?
I would give a slightly more direct explanation than WhatsUp. The Poisson summation formula states that for $f$ a smooth rapidly decreasing function on $\mathbb R$, $\hat{f}$ the Fourier transform, ...
- 148k
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. ...
- 91.8k
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 ...
- 47.4k
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 ...
- 16.2k
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\...
- 188k
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}\...
- 20.4k
6
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 ...
- 91.8k
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 ...
- 91.8k
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 \...
- 20.8k
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 ...
- 24.2k
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
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^{\...
- 66
5
votes
Accepted
$q$-analog of an integral from quantum field theory?
The function
$$
f_q(x,y,z)=\sum_{cyc}e^z\frac{\theta_q\left(e^{\frac{2 \pi i}{3}+x-z}\right) \theta_q\left(e^{\frac{2 \pi i}{3}+y-z}\right)}{\theta_q\left(e^{x-z}\right) \theta_q\left(e^{y-z}\right)}...
- 5,624
5
votes
Siegel modular forms in Mathematica
You might want to check out the documentation on Siegel Modular Forms by Yuen, Poor, Shurman, and King. It provides a variety of Mathematica and Maple notebooks.
- 188k
5
votes
Accepted
Is the left derivative of this theta function zero at $1$?
Writing $x=e^{-a}$ for $a>0$ and then using the Poisson summation formula with $s(u)=e^{i\pi u} u^2 e^{-au^2}$ (in the notation of the Wikipedia article), we rewrite the sum in question as
$$G(a):=\...
- 128k
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 $-...
- 7,239
4
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, ...
- 3,927
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 ...
- 91.8k
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 $9$,
$$ \frac ac=a^3-b^3,\quad\...
- 2,784
4
votes
Accepted
2D lattice sum with numerator
The $n=4$ sum is accessible as follows. Expand $(j+k)^4$; by symmetry
the odd terms $4j^3k$ and $4jk^3$ cancel out so we need only sum
$(j^4 + 6j^2k^2 + k^4) / (j^2+k^2)^4$. You say you already know
...
- 79.9k
3
votes
Infinite product of $1-q^{n^2}$
Write $\theta(q)=\sum_{n \in \mathbb Z} q^{n^2}$. Taking logarithms, we get that the logarithm of your function is
$\sum_n \ln(1-q^{n^2}) = -\sum_n \sum_m \frac 1 m q^{n^2m} = -\sum_m \frac 1 m \frac {...
- 3,738
3
votes
Obtain a series expansion of $a^2(q)a^2(q^4)$
The same group of authors work out the answer to your question in theorem 2 of this paper, so you can simply cite their result.
As mentioned in the comments, the answer gets more complicated because ...
- 85.5k
3
votes
Accepted
Semidirect product of metaplectic group and Heisenberg group
I presume you are looking for a faithful action of $Mp_{2n}$ on something related to the Heisenberg group $H_{2n+1}$. This is well-known as Weil Representation.
In the modern language, consider $Mp_{...
- 12.3k
3
votes
" Laurent expansion" of quasi-periodic complex complex function
Bloch's theorem says that $f(z)=e^{iθz/a}u(z)$, with $u(z+a)=u(z)$. Then $u(z)=\sum_{n=-\infty}^\infty c_n \xi^n$ has a series expansion in powers of $\xi=e^{2\pi iz/a}$, which can be seen as a ...
- 188k
3
votes
Accepted
"Sparse" Theta Series
Won't the following argument show that the difference between successive exponents can never be bounded away from zero no matter how clever you try to be in selecting $(a,b)$?
The idea is to consider ...
- 1,362
3
votes
Accepted
On $x^k+y^k=1$ and the Dixonian elliptic functions
Define the generalized trigonometic functions (discussion of these functions is given in this answer)
$$
z=\int_0^{\sin_{pr}z}\frac{dt}{\sqrt[p]{1-t^r}},\qquad \cos_{pr}z=\sqrt[r]{1-(\sin_{pr}z)^r},\...
- 5,624
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