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

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3
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0answers
48 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= ...
0
votes
0answers
34 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 ...
1
vote
0answers
30 views

Elliptic formulation of sum with cosine and hyperbolic cosine

Can anyone check my reformulation below?$$\sum_{k\in \mathbb{Z}}\frac{1}{\cosh{(k+z)\pi}+\cos{(k+z)\pi}}=\frac 2\pi \csc{\pi ...
2
votes
1answer
90 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, ...
3
votes
1answer
169 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 ...
0
votes
0answers
25 views

the zeros of theta function?

Recall that the theta function with character $(a,b)\in \mathbb{R}^2$ is defined by $$ \vartheta_{a,b}(z, \tau) :=\sum^\infty_{n=-\infty} e^{\pi i (n + a)^{2} \tau + 2 \pi i(n + a)(z + b)},\quad a,b ...
3
votes
1answer
54 views

Some analogs of the pentagonal number theorem

There are the following analogs of the famous identity $$ \prod_{n\geqslant1}(1-q^n)=\sum_{n\in\mathbb Z}(-1)^nq^{\frac{3n^2-n}2}. $$ Let $v_2(n)$ denote the 2-adic valuation of $n$, that is, the ...
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0answers
53 views

Elliptic theta function approximations

I have encountered the formulas, which are approximations of Elliptic Theta functions, $$\vartheta _4\left(0,\frac{1}{\sqrt{e}}\right)\simeq 2 \sqrt{2 \pi } \sqrt[8]{\left(4+\cosh \left(4 \pi ...
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vote
0answers
24 views

Holomorphicity and modularity of Jacobi forms

I am reading this paper Suerconformal algebras and mock theta functions by T. Eguchi and K. Hikami. In section 3.2, the authors define a function $$J(z;w;\tau) = ...
4
votes
1answer
63 views

Jacobi Elliptic Functions built from Jacobi theta functions

I believe I understand the general theory of elliptic functions to an extent. What I can't seem to find is the distinct method which was used to show that a particular combination of Jacobi Theta ...
0
votes
0answers
33 views

Reciprocals of theta functions

I've spent the last few months with partial fraction expansions, and thought to create a function with simple poles over a lattice of zeros, like that of any of the Jacobi theta functions... but I ...
1
vote
1answer
21 views

any interpretation for $\left[ \sum_{n\in \mathbb{Z}} q^{n^2} w^{2n} \right]^{-1}$?

One very simple version of the theta function is as a generating function over the perfect squares: $$ \theta(\tau; z) = \sum_{n\in \mathbb{Z}} q^{n^2} w^{2n} $$ Where $q = e^{2\pi i \tau}$ and $w = ...
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votes
0answers
13 views

Is the $z\gg 1$ behavior of the theta function $\theta_1(z;q)$ known?

Is the $z\gg 1$ behavior of the theta function $\theta_1(z;q)$ known? It seems naively like it could be estimated by a Gaussian integral which would give an answer along the lines of $e^{bz^2}$ for ...
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0answers
30 views

Which theta function is $\theta(x;q) = (x;q)(q/x;q)$?

The physics paper I am reading very non-chalantly defines the theta function as $$ \theta(x;q) = (x;q)(q/x;q) \hspace{0.5in} \tilde{\theta}(x;q) = x^{-1/2}(x;q)(q/x;q) $$ where they are using the ...
2
votes
0answers
40 views

Fibonacci-related infinite sum

Prove that $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}}=\frac{\sqrt{5}}{4}\theta_2^2\bigg(\frac{3-\sqrt{5}}{2}\bigg)$$ and $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}+F_{2k-1}}=\frac{(2k-1)\sqrt{5}}{2F_{2k-1}}$$ ...
2
votes
0answers
23 views

Equation of the curve corresponding to a principal polarization

Let $\mathbb{C}^2/\Lambda$ be a principally polarized abelian surface. I think it is well-known how to write down the equation of the divisor (Riemann surface) corresponding to the polarization, in ...
1
vote
1answer
45 views

Is there a limiting case for this sequence of infinite product representations for the theta function?

Starting from the famous infinite product $$ (1+z)^2(1-z^2)(1+z^3)^2(1-z^4)(1+z^5)^2(1-z^6)\cdots=1+2z+2z^4+2z^9+2z^{16}+\dots $$ it is easy to show by induction that $$ ...
0
votes
1answer
86 views

A special case of triple product identity

Can any one guide me how can I prove these identities?? $$\sum_{n=-\infty}^\infty (-1)^nq^{(6n+1)^2}=q \prod_{n=1}^\infty(1-q^{24n})$$ $$\sum_{n=-\infty}^\infty (4n+1)q^{(4n+1)^2}=q ...
0
votes
0answers
39 views

Zeroes of Jacobi Theta Functions

Based on wolfram alpha: $$\sum_{i=0}^{\infty}[x^{i^2}] = \frac{1}{2}(v_3(0,x) +1) $$ Whereas $v_3$ is the third Jacobi Theta function. See: http://bit.ly/1FIyUTq I am curious for what values in ...
2
votes
0answers
22 views

Does this function appear in theory of modular forms?

Does the following function $\phi(q,z)$ appear in theory of modular forms $$ \phi(q,z):=\prod_{n=-\infty}^{\infty}(1+q^nz) $$ for $(q,z)$ in some domain in $\mathbb{C}^2$? Any reference will be ...
0
votes
1answer
32 views

Mumford's proof of theta function convergence

On Dave Mumford's Tate Lectures on Theta I, he begins by proving that $\theta(z,\tau)$ converges. It begins something like: Let $|Im(z)|<c$ and $Im(\tau)>\epsilon$, then: $|e^{\pi i n^2 ...
2
votes
1answer
43 views

Derivation of Transformation for basic theta function

Given that $\vartheta(x) = \sum_{n = -\infty}^\infty e^{-\pi n^2 x}$, I am trying to finish a derivation that $\vartheta(x) = \frac{1}{\sqrt{x}}\vartheta(1/x)$. I believe that I am very close. I ...
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0answers
24 views

seeking upper/lower bounds of a function $F(m)$ related to Jacobi theta function

I am looking for the upper/lower bounds of function $F(m)$ defined and plotted above. The function is related to Jacobi theta function $\theta(x)$ and its derivative values at $x=1$: ...
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0answers
38 views

What is this quotient of partial derivatives of the first Jacobi's Theta function

Let $\theta_1(x,\tau)$ be the first Jacobi's theta function: $$ \theta_1(x)=\theta_1(x,\tau)=-i \sum_{n\in \mathbb Z} (-1)^{n}e^{i \pi (n+1/2)^2\tau}e^{2i\pi(n+1/2)x}\qquad x\in \mathbb C, \, \tau\in ...
1
vote
2answers
50 views

Ellptic\Jacobi theta function and its residue integral

The Ellptic\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}) \\ &= ...
1
vote
0answers
60 views

q-theta function and their properties

I want to compute the residue integral for the $q$-theta function, and derive its properties. First, I'll briefly explain the definition \begin{align} & ...
3
votes
0answers
57 views

Freeman Dyson's identity for the modular discriminant $\Delta$

In his 'Introduction to Modular Forms', Don Zagier states the Freeman Dyson's identity : $$\Delta(\tau)=\sum_{\substack{ (x_1,\ldots,x_5)\in \mathbb{Z}^5 \\ x_1+\cdots+x_5=0 \\ x_i \equiv i ...
2
votes
0answers
54 views

When are theta constants modular

I'm looking at $\theta$ constants with characteristic, defined by $$\theta\left(\begin{array}{c} \epsilon \\ \epsilon' \end{array}\right)(z,\tau) = \sum_{n\in \mathbb{Z}} \exp 2\pi i \left\{ ...
2
votes
0answers
44 views

Motivation behind theta function calculation in Diamond & Shurman

I am reading section 1.2 of A First Course in Modular Forms. Let $q=e^{2\pi i\tau}$, where $\tau\in\cal H$ is assumed to be in the upper half plane, and define $\theta(\tau)=\sum_{n\in\Bbb Z}q^{n^2}$. ...
2
votes
1answer
78 views

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 ...
3
votes
1answer
39 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, ...
2
votes
1answer
76 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
votes
0answers
71 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 ...
3
votes
0answers
66 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 ...
3
votes
0answers
87 views

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 ...
3
votes
2answers
158 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) = ...
13
votes
1answer
258 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 ...
0
votes
0answers
65 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 ...
2
votes
0answers
22 views

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 + ...
5
votes
1answer
118 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 ...
8
votes
0answers
197 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 ...
1
vote
1answer
68 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 } ...
4
votes
1answer
160 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, ...
1
vote
1answer
163 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 ...
4
votes
2answers
370 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
votes
1answer
125 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( ...
7
votes
0answers
334 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 ...
1
vote
1answer
116 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
votes
1answer
124 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} < ...
7
votes
1answer
101 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 ...