For questions about $\theta$ functions (special functions of several complex variables).

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7
votes
2answers
342 views

Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some ...
8
votes
2answers
161 views

the ratio of jacobi theta functions and a new conjectured q-continued fraction

Given the squared nome $q=e^{2i\pi\tau}$ with $|q|\lt1$, define $$\begin{aligned}H(q)=\cfrac{2(1+q^2)}{1-q+\cfrac{(1+q)(1+q^3)}{1-q^3+\cfrac{2q^2(1+q^4)}{1-q^5+\cfrac{q^3(1+q)(1+q^5)}{1-q^7+\cfrac{q^...
8
votes
2answers
238 views

series involving $\log \left(\tanh\frac{\pi k}{2} \right)$

I found an interesting series $$\sum_{k=1}^\infty \log \left(\tanh \frac{\pi k}{2} \right)=\log(\vartheta_4(e^{-\pi}))=\log \left(\frac{\pi^{\frac{1}{4}}}{2^{\frac{1}{4}}\Gamma \left( \frac{3}{4}\...
8
votes
1answer
1k views

On the growth of the Jacobi theta function

So, I ran into this exercise from Stein & Shakarchi. CA, Chapter 5: Show that if $\tau$ is fixed with positive imaginary part, then the Jacobi theta function $$\theta(z | r) = \sum_{n=-\infty}^{\...
11
votes
1answer
674 views

Hermite's solution of the general quintic in terms of theta functions

Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
5
votes
1answer
131 views

Rogers-Ramanujan continued fraction in terms of theta functions?

The Rogers-Ramanujan cfrac is, $$r = r(\tau)= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$ If $q = \exp(2\pi i \tau)$, then it is known that, $$\frac{1}{r}-r =\frac{\eta(\tau/5)}{\eta(5\...
5
votes
2answers
494 views

Convergence to $0$ of Jacobi theta function

I'm trying to prove that a function $$f(y) = \sum_{k=-\infty}^{+\infty}{(-1)^ke^{-k^2y}}$$ is $O(y)$ while $y$ tends to $+0$. I have observed that $f(y) = \vartheta(0.5,\frac{iy}{\Pi})$ where $\...
14
votes
1answer
298 views

Proving $\left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$

The so-called "two squares theorem" can be proven by establishing the following identity: $$\left(\sum_{n=-\infty}^\infty e^{\pi i \tau n^2}\right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \...
7
votes
1answer
303 views

How to solve the general sextic equation with Kampé de Fériet functions?

It is frequently stated, for example on Wolfram Mathworld, that the general sextic equation $$x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 = 0$$ can be solved in terms of Kampé de ...
6
votes
2answers
181 views

A $q$-continued fraction connected to the divisor function?

In this post, the following two continued fractions discussed by Nicco are given, $$A(q)= \left(\frac{\vartheta_2(0,q)}{2\,q^{1/4}}\right)^2= \cfrac{1}{1-q+\cfrac{q(1\color{red}-q)^2}{1-q^3+\cfrac{q^...
5
votes
1answer
167 views

Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{n}\left(e^{-\pi x}\right)}{1+x^2} dx $

The Jacobi theta function $\theta_4$ is defined by $$\displaystyle \theta_4(q)=\sum_{n \in \mathbb{Z}} (-1)^n q^{n^2} \tag{1}$$ For this question, set $q=\large e^{-\pi x}$ and $\theta_4 \equiv \...
3
votes
0answers
99 views

conjectured identity of the product of two theta functions

Looking into the discussion in this post,I was naturally led to consider the following general identity Given the two jacobi theta functions,$$\theta_2(q)=\sum_{n=-\infty}^\infty q^{(n+1/2)^2}$$ and $...
1
vote
0answers
57 views

proof of jacobi triple product using ramanujan's notation

Inspired by this proof in mathworld,I rewrote the proof in terms of ramanujan theta function Define the function $$M(c)=\prod_{n = 1}^{\infty}(1 +a^{n}b^{n-1}c)(1+\frac{a^{n-1}b^{n}}{c})\tag1$$ then ...
9
votes
0answers
361 views

Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for ...
2
votes
1answer
105 views

What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$

If $\vartheta_{2}(q)$ is jacobi's theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, ...