Due to this question, I found myself trying to take the following integral: $$\int_0^x\vartheta_3(0,t)\ dt=\ ?$$ However, I know not of how to do this. As per this post, I find the evaluation at $x=1$ to be $\frac\pi{\tanh\pi}$. It is equivalent to trying to evaluate the following series: $$\sum_{n=0}^\infty\frac{x^{n^2}}{n^2+1}$$ My end problem is that I […]

In this paper, it says that the three Borwein cubic theta functions obey the identity $a(q)^{3}=b(q)^{3}+c(q)^{3}$, which is strongly reminiscent of the identity that Dixonian elliptic functions obey $\mathrm{sm}^{3}(z)+\mathrm{cm}^{3}(z)=1$. What relationship (if any) exists between the Dixonian elliptic functions and the Borwein cubic theta functions?

This question already has an answer here: Proving $\left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$ 1 answer

What exactly is a theta function $$\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)= 1 + 2 \sum_{n=1}^\infty \left(e^{\pi i\tau}\right)^{n^2} \cos(2\pi n z) = \sum_{n=-\infty}^\infty q^{n^2}\eta^n”$$ and how does it arise naturally? (as in how does it just fall out of a computation illustrating the necessity for taking […]

The Borwein cubic theta functions $a(q),b(q),c(q)$ and the doubly-periodic Dixonian elliptic functions $\mathrm{sm}(z), \mathrm{cm}(z)$ parameterize the Fermat cubic $x^3+y^3=1$, so $$\frac{b^3(q)}{a^3(q)}+\frac{c^3(q)}{a^3(q)} =1\tag1$$ $$\mathrm{sm}^{3}(z)+\mathrm{cm}^{3}(z)=1\tag2$$ This post asks if there is any relation between the Borwein and Dixonian versions. It is known that the former is associated with $_2F_1\big(\tfrac13,\tfrac23;\color{blue}1;u\big)$ while the latter is with $_2F_1\big(\tfrac13,\tfrac23;\color{blue}{\tfrac43};v\big)$. For argument […]

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 $\vartheta$ is a Jacobi theta function. It seems that these functions are very well studied but I am not too familiar with this area. Any useful links or suggestions are […]

There are various identities for the Jacobi Theta Functions $\vartheta_n(z,q)$ on the MathWorld page and on the Wikipedia page. But I found no integral identities for these functions. Meanwhile, there are beautiful identities for the simple case of $z=0$: $$\int_0^1 \vartheta_2(0,q)dq=\pi \tanh \pi$$ $$\int_0^1 \vartheta_3(0,q)dq=\frac{\pi}{ \tanh \pi}$$ $$\int_0^1 \vartheta_4(0,q)dq=\frac{\pi}{ \sinh \pi}$$ I found these identities […]

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, $$\frac{1}{r}-r =\frac{\eta(\tau/5)}{\eta(5\tau)}+1\tag1$$ $$\frac{1}{r^5}-r^5 =\left(\frac{\eta(\tau)}{\eta(5\tau)}\right)^6+11\tag2$$ with the Dedekind eta function, $\eta(\tau)$. Q: Is there a similar simple identity known using ratios of the Jacobi theta functions $\vartheta_n(0,q)$? $\color{brown}{Edit}$: In response to a comment, here are some details. […]

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 page for the quintic, but I’m interested in Hermite/Kronecker’s process/proof)

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 $\theta_4 \equiv \theta_4(q)$. Define $\theta_3(q)=\theta_4(-q)$. Using Lambert-Series representation for powers of $\theta_4$ (which I will describe in a moment) and integrating term by term, I have obtained a family of neat identites: […]

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