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

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About Jacobi Theta function

The Jacobi theta function is given by \begin{align} \Theta_1(\tau|z)&=\Theta_1(q,y)=-iq^{\frac{1}{8}}y^{\frac{1}{2}}\prod_{k=1}^{\infty}(1-q^k)(1-yq^k)(1-y^{-1}q^{k-1}) \\ &= -i\sum_{n\in ...
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19 views

Injectivity of Theta functions

Let $\vartheta_{00}$ and $\vartheta_{01}$ be Jacobian Theta functions (notations like on wikipedia). $F:=\left\{ \tau \in \mathbb{C}: Im(\tau)>0, \left| Re(\tau)\right|<1, ...
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1answer
51 views

Conclude behaviour of holomorphic function on interior from behaviour on boundary - by the example of Theta function

Actual Problem I feel like by attempting to transfer my problem to a more general form it suffered a lot of important detail. So here i present to you my actual problem. It's part of a proof in an ...
2
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0answers
46 views

Theta-divisor and special divisors

I'm studying the Jacobi inversion problem. Could you help me with the next question? Let us consider an algebraic curve $C$ and it's Jacobian $J(C)$. For a initial point $P_0$ we can construct the ...
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0answers
34 views

Values of derivatives of Jacobi theta function

The Jacobi theta function is defined as: $$\theta(x)=\sum_{n=-\infty}^{\infty}\exp(-\pi n^2 x)\text{ }, x>0$$ On wikipedia.org, we can find close-form expressions for the values of ...
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Modular forms on the theta group

The theta group $\Gamma$ is the subgroup of $SL(2;\mathbb{Z})$ generated by $T=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$ It ...
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1answer
67 views

What is a Theta Function?

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) = ...
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1answer
203 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 ...
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49 views

Bound of integral involving theta function

I need to prove that the following bound is true. I thought this might follow from the inversion property of the theta function, as the infinite sum in the integrand is precisely ...
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Is it true that $\theta_{1,1}^{4N} \in J_{2N,2N}(2N)$?

I need examples of Jacobi forms for full congruence subgroups $\Gamma(N) $ of $SL(2,Z)$. As a particular case, take the theta function $\theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + ...
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1answer
93 views

The “$\mathbb{Z}_n$-theta function” - what is it? Is it being studied somewhere?

The Jacobi theta function is well known: $$\theta(z, \tau) = \sum_{n=-\infty}^\infty \mathrm{e}^{\pi i n^2 \tau + 2\pi i nz}$$ In Shahn Majid's "Foundations of Quantum Group Theory", you'll find a ...
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148 views

Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
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1answer
52 views

Expansion of Jacobi's $\theta_3(0,q)$ in q=1

In trying to solve a certain limit, I wondered how Mathematica comes up with this weird expression for a series expansion of Jacobi's $\theta_3(0,q)$ in $q=1$ at the order 0: $\frac{i \sqrt{\pi } ...
3
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1answer
134 views

Monstrous Moonshine for $M_{24}$?

This is connected to my MO post "Monstrous Moonshine for $M_{24}$ and K3?". In page 44 of this paper, eqn(7.16) and (7.19) yield, ...
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1answer
116 views

Theta function: Absolute convergence

I have to proof that the series $$ \theta (z)= \sum_{n=-\infty}^{+\infty} e^{\pi i [n^{2} \tau + 2nz]}$$ converges absolutely and uniformly on compact subsets of $\mathbb{C}$. Necessary and sufficient ...
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2answers
183 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 ...
4
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1answer
117 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( ...
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297 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 ...
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1answer
95 views

Coefficients of powers of the theta function

Let $q=\exp(2 \pi i z)$ and $$\theta(z)=\sum_{n=-\infty}^\infty q^{n^2}.$$ Now, I shall show that the powers of $\theta$ are given by $$\theta(z)^r = \sum_{n=0}^\infty S_r(n) q^n$$ where $S_r(n)$ ...
2
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1answer
93 views

Jacobi Theta function inequality

I am trying to show that $$\sum_{n=1}^\infty e^{- \pi n^2 x} < \frac{1}{2} x^{-\frac{1}{2}}, \; \forall x>1$$ Here's what I'm doing: $$\sum_{n=1}^\infty e^{- \pi n^2 x} < ...
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1answer
94 views

Generating function for $r^\binom{n}{2}$

I'm trying to find a closed form of the generating function $$ G(x) = \sum_{n \ge 0} r^\binom{n}{2} x^n $$ for a real number $0 < r < 1$. I found that $G(x) = 1 + xG(rx)$. Any hints where to ...
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72 views

Interpretation of functional equation of dedekind eta function

It's well known that the Dedekind eta function defined by $\eta(z) = \displaystyle e^{\frac{\pi i z}{12}} \prod_{n=1}^{\infty} (1 - e^{2 \pi i n z}) $ converges for $z$ in the upper half plane to a ...
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Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)

I need to evaluate this series for arbitrary $\beta > 0$: $ Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $ Is it related to a known transcendental function? From the research I did, it ...
3
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1answer
350 views

Calculating a summation of a $\theta$ function

Let $ \theta_z(t) = \sum \limits_{m,n\in\mathbb{z}}e^{-\pi Q_z(m,n)t}$ where $Q_z(m,n)=y^{-1}|mz+n|^2$. I need to prove that $\theta_z(t)=t^{-1}\theta_z(t^{-1})$. Now, looking that up I know that ...
13
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1answer
281 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
13
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2answers
298 views

Combinatorial interpretation of this identity of Gauss?

Gauss came up with some bizarre identities, namely $$ \sum_{n\in\mathbb{Z}}(-1)^nq^{n^2}=\prod_{k\geq 1}\frac{1-q^k}{1+q^k}. $$ How can this be interpreted combinatorially? It strikes me as being ...
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149 views

A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+… = 0$

Given the Dedekind eta function $\eta(\tau)$. Define, $y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$ Prove the multi-grade identity [1], $y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + ...
5
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0answers
207 views

How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
7
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1answer
780 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) = ...
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167 views

What is meant by a set of generic points on a compact Riemann surface

Let $X$ be a compact connected Riemann surface of genus $g \geq 1$. I'm studying a theorem of Faltings which looks as follows. Let $P_1,\ldots, P_g$ be generic points on $X$. Then we have some ...
4
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1answer
107 views

Reference request: theta functions for lattices which are not unimodular or even

In A Course in Arithmetic, Serre works out some of the theory of theta functions for even, self-dual lattices, e.g. such theta functions are modular forms of weight $n/2$, where $n$ is the dimension ...
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127 views

Relationship between Dixonian elliptic functions and Borwein cubic theta functions

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 ...
11
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1answer
515 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
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1answer
350 views

Where are this kind of series used, $\vartheta_{4}(0,e^{\alpha \cdot z})$?

In my recent explorations I stumbled upon the following series $$ \vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z ...