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

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5
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1answer
51 views

Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$

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, ...
2
votes
1answer
86 views

Theta series and Jacobi theta functions

I have some difficulties with expressing the following series $\sum\limits_{b=-\infty}^{+\infty}\sum\limits_{a=-\infty}^{+\infty}q^{1 + 3 a + 3 a^2 - 3 b - 3 a b + 3 b^2}$ using standart theta ...
4
votes
0answers
31 views

Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$ \theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) , $$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
4
votes
1answer
90 views

$E_8$ and theta functions

The root lattice $\Gamma_8$ of the exceptional Lie algebra $E_8$ is an eight-dimensional lattice which consists of lattice points in $\mathbb{R}^8$ which with respect to an orthonormal basis $e_1, \...
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0answers
37 views

General Form of Theta Functions from Functional Equations

From Elliptic Curves: Function Theory, Geometry and Arithmetic by McKean and Moll: Exercise 3.1.2. Discuss the general solution of the two identities (a) $f(x+2)=f(x)$ and (b) $f(x+2\omega)=e^{ax+b}f(...
15
votes
1answer
174 views

Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following ...
2
votes
1answer
70 views

Prove that $\prod_{k=1}^{\infty}\left(1-e^{-\pi{k\sqrt n}}\right)=\frac{\sqrt2 g_n^2\psi(e^{-\pi\sqrt n})}{e^{\frac{\pi\sqrt n}{12}}}$

The source of idea is from here Where $$\psi(q)=\sum_{k=0}^{\infty}q^{\frac{k(k+1)}{2}}$$ Prove that, (1) $$\prod_{k=1}^{\infty}\left(1-e^{-\pi{k\sqrt n}}\right)=\frac{\sqrt2 g_n^2\psi(e^{-\pi\...
5
votes
1answer
189 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 \...
2
votes
0answers
41 views

Implementation of Jacobi theta functions in Matlab

I need Jacobi theta functions for my Matlab program. The functions are not included in the predefined Matlab functions. Doing a simple Google search, I found a package developed by Moiseev I. in 2008. ...
10
votes
1answer
323 views

How to estimate a specific infinite sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...
2
votes
0answers
24 views

show that $\sum P(x,y) e^{x^2 + y^2}$ is a modular form over $\Gamma_0(4)$ where $P(x,y) = x^4 - 6 x^2 y^2 + y^4$

In these lecture notes of Zagier, I read that generalized theta functions are still modular forms. Let $q = e^{2\pi i z}$ $$\theta(z) = \sum_{(x,y) \in \mathbb{Z}} \Big[ x^4 - 6 x^2 y^2 + y^4 \Big] \...
0
votes
0answers
22 views

Let $\theta(z) = \sum q^{n^2}$, is $\theta(-1/z)$ also a theta function?

I am learning about theta functions. Let $q = e^{2\pi i \, z}$ and $\theta(z) = \sum q^{n^2}$. How does it behave under $\mathrm{SL}_2(\mathbb{Z})$ ? In general we have: $$ \theta\left( - \frac{1}...
1
vote
0answers
61 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 ...
2
votes
0answers
58 views

Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
3
votes
0answers
36 views

Degree of the Divisor of a Theta Function

Let $(\gamma_1, \gamma_2)$ be a base for a lattice $\Gamma$ in $\mathbb C$, and $\theta$ a theta function, ie an holomorfic function such that $\theta(z+ \gamma) = \theta(z)e^{2i\pi(a_\gamma z + b_{\...
3
votes
0answers
59 views

Modular Discriminant and Pentagonal Numbers

I am asked to show $$(2\pi)^{-12}\Delta(\tau) = q \cdot \Big (\sum_{n\in \mathbb{Z}} (-1)^n \cdot q^{(3n^2+n)/2} \Big)^{24}$$ where $\Delta:\mathbb{H} \to \mathbb{C}$ is the modular discriminant, $q=e^...
4
votes
1answer
142 views

About the sums $\sum_{n=1}^\infty x^{n^2}$ and $\sum_{n=1}^\infty \frac{x^n}{1+x^{2n}}$

Despite all my efforts trying to crack these, i haven't been able to do so. An approach that i've tried gives me somewhat of an asymptotic approximation, but still fails to produce the values near x=0....
2
votes
1answer
40 views

Why is this not a complex variable?

I'm watching a video about the Jacobi Theta Function on YouTube. At around 6:14 in the video, he has shown that... $$\vartheta (x) = \sum_{n\in \mathbb Z} e^{-\pi {n^2} x} = \sum_{k\in \mathbb Z} {e^...
2
votes
1answer
58 views

Expression for Jacobi theta function derivative

Please explain how to get $$\frac{\vartheta'_{1}(z, q)}{\vartheta_{1}(z, q)} = \cot z + 4\sum_{n = 1}^{\infty}\frac{q^{2n}}{1 - q^{2n}}\sin 2nz$$ where $\vartheta_{1}(z, q)$ is one of the Jacobi theta ...
3
votes
1answer
77 views

series involving $\log(\tanh(\pi k/2))$ II

continuation of the question above series involving $\log\tanh(\pi k/2)$) it is posible to prove that $$\sum_{k=1}^\infty \log (\tanh (k x))=\sum _{n=1}^\infty \left(\frac{\tanh \left(\frac{\pi ^2 n}{...
1
vote
1answer
33 views

Show that $f(qz) = qz^2 f(z)$ where $f(z) = [\theta(z) \theta(1/z)]^{-1}$

Let $\theta(z) = (z;q)(q/z; q)$ where $(q;z) = \prod_{i=0}^\infty (1 - zq^i) $. Then let $f(z)$ be defined by: $$ f(z) = \frac{1}{\theta(z) \theta(1/z)} $$ Show that $\boxed{\color{#0033FF}{f(qz) = ...
3
votes
0answers
68 views

What is the series expansion of reciprocal of theta function $\frac{1}{\theta(z;q)}$?

"The" theta function is an ambiguous concept, but one definition I have found is: $$ \theta(z;q) = (z;q)_\infty(q/z;q)_\infty = \frac{1}{(q;q)_\infty}\sum_{k \in \mathbb{Z}}z^k q^{\binom{k}{2}} \tag{$...
4
votes
1answer
81 views

Jacobi theta with a matrix

I would like to evaluate $$ \sum_{q_1 = -\infty}^{\infty} \cdots \sum_{q_N = -\infty}^{\infty} e^{-\sum_{j}\sum_{k} q_{k} A_{kj} q_{j}} $$ with $A$ a real $N\times N$ symmetric matrix. I know how to ...
-1
votes
1answer
45 views

What is theta value?

Using a pre-defined formula in Desmos android app the following example is given : What is value of theta used within formula for evaluating r ? Is it some implicit value, as it's value is not ...
2
votes
0answers
47 views

What is known about $\sum_{n=0}^{\infty} x^{n^3} $.

$f(x) =\sum_{n=0}^{\infty} x^{n^2}$ and similar "theta-type" functions are extensively studied. They have many properties and occur in number theory , algebra (in particular solving the quintic ...
2
votes
0answers
37 views

Zeros of the derivative of a theta function

I'm trying to find the positions or at least the number of zeros of $\partial_z\vartheta[p,0](z,\tau)$ where the standard theta function with characteristics $(p,q)$ is defined as $\vartheta[p,q](z,\...
0
votes
1answer
74 views

Evaluating: $ \sum_{n=0}^{\infty}x^{n^{2}} $

How do you evaluate: $\displaystyle \sum_{n=0}^{\infty}x^{n^{2}} $ Or more generally $ \large\displaystyle \sum_{n=0}^{\infty}x^{n^{\alpha}} $ Note that: $|x| <1$
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vote
0answers
26 views

Coefficients of a power series, an algebraic identity

Given the following 4 functions: \begin{align} \theta_1(0, \tau) &= 2(q^{1/4} + q^{9/4} + q^{25/4} + \ddots), \\ \theta_2(0, \tau) &= 1-2q+2q^4-2q^9+ \ddots), \\ \theta_3(0, \tau) &= 1+2q+...
2
votes
3answers
83 views

How fast does $\Sigma_{n=1}^{\infty} (1- \exp(-x))^{n^2} $ grow?

Let $x$ be a positive real. Define $\theta(x) = \Sigma_{n=1}^{\infty} (1- \exp(-x))^{n^2} $. How fast does $\theta(x)$ grow ? In other words what is a good asymptotic for it when $x$ is large ? Maybe ...
3
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0answers
47 views

Is $\sum_n\exp(ian+ibn^2+icn^3)$ known in terms of anything else?

For arbitrary $a,b,c$, does the series $$F(a,b,c)=\sum_{n=-\infty}^\infty\exp\left(ian+ibn^2+icn^3\right),$$ i.e. an evenly-weighed series of exponentials of cubic polynomials, converge to anything ...
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0answers
28 views

Summation of an infinite q series

When calculating a Partition function, I encounter the following summation $$\sum_{n=0}^{\infty} x^n q^{n^2}.$$ I know that the sum$\sum_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function , but I ...
7
votes
1answer
143 views

How do I show Theta function is not identically zero

How do I show that for a fixed $\omega$ in the upper half plane, the theta function $\theta(z) = \sum_{n=-\infty}^{\infty} e^{\pi i(n^2\omega + 2nz)}$ is not identically zero? Is there an obvious ...
8
votes
2answers
162 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^...
2
votes
0answers
77 views

Jacobi Theta Functions?

For the Jacobi theta function $\vartheta_3(z|\tau)$ there exists an equality (by Whittaker & Watson) \begin{equation} \vartheta_3(z|\tau) = \sum_{n=-\infty}^{\infty} e^{n^2 \pi i \tau + 2 n i z} =...
3
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0answers
101 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 $...
5
votes
1answer
132 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\...
8
votes
1answer
156 views

What is a mock theta function?

We define a mock theta function as follows: A mock theta function is a function defined by a $q$-series convergent when $|q|<1$ for which we can calculate asymptotic formulae when $q$ tends ...
4
votes
2answers
122 views

Proof of Functional Equation Zeta

$$ \pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-(1-s)/2}\zeta(1-s)\Gamma((1-s)/2) $$ (That's the equation that want to prove) Hello guys, so I'm trying to prove the functional equation of Riemann Zeta, ...
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^...
7
votes
1answer
319 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 ...
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0answers
56 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 z}D_z\left[\tan^{-1}\left(\frac{\theta(z/2,2)}{\vartheta(z/2,2)}\right)\...
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, ...
7
votes
2answers
347 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
1answer
94 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 \...
4
votes
1answer
74 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 ...
1
vote
0answers
79 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 ^2\right)...
2
votes
0answers
44 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) = \frac{(\theta_{11}(z;\tau))^2}{(\eta(\...
4
votes
1answer
110 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
1answer
83 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 can'...
2
votes
1answer
29 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 = ...