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Questions tagged [theta-functions]

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

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Closed form of $\sum_{n=1}^\infty q^{- n^2} z^n$

In this question the summation goes from $-\infty$ to $\infty$ and the answer has a pretty "good" closed form. Now I came across the sum $\sum_{n=1}^\infty q^{-n^2} z^n$ where $|z|<1$ and I don't ...
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From root and weight lattices of SU(N) to $\theta$-functions as sections of a line bundle and $CP$-space

I have troubles to digest the following messages/discussions in the following work in p.10-12; Which construct a map from the moduli space of flat connections $M_{\rm flat}=\mathbb{E} / {\mathfrak S}...
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Finding doubly periodic solutions to partial differential equations

Say I have a certain PDE in real variables $x$ and $y,$ which might be nonlinear, so that we can't necessarily just throw a Fourier series at it. By way of some intuition, let's say, I have a very ...
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1answer
40 views

Transformation of a theta function

Given $\Theta(\tau)=\sum_{n \in \mathbb Z}exp(2\pi in² \tau)$ and $\tau \in \mathbb H$ I am trying to prove the following identity: $\Theta(-\frac{1}{2\pi})=\sqrt{\frac{\tau}{i}}\Theta(\frac{\tau}{2}...
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A relation concerning the “sum of squares” counting function $r_2(n)$

Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here the sign as well as order of summands matters. Also by convention we ...
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Evaluation of integral of multiple Jacobi theta functions

This problem is related to the Fermi gas problem of quantum mechanics. Define $\rho(x,x')$ as follows $$ \rho(x,x') = \theta_4(x) \frac{\theta_1^2(x/2)}{\theta_4^2(x/2)} \frac{\theta_2(\frac{x-x'}{2}...
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How many integer solutions are there to $x^4+y^4-x^2y^2=n$. Is there a generating function for this?

It would be kind of cool to get a closed form for the number of integer solutions $$x^4+y^4-x^2y^2=n$$ which we will let $\phi_n$ denote. It would be cool because we could exploit $\sum_{n=1}^N\...
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36 views

Theta function equation

I have been trying to prove an equation of a theta function. I understood that it some how related to the Poisson summation formula, but no luck. Any help would be appreciated. the exercise
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Explicit Values of the Jacobi Theta Function [duplicate]

The sum $\sum_{n=-\infty}^{\infty}\exp(-\pi n^2) $, or $\vartheta(0;i)$ (Jacobi Theta Function) has a closed form solution of $\frac{\pi^{\frac{1}{4}}}{\Gamma(\frac{3}{4})}$ but nowhere have I been ...
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Evaluation of the limit $\lim_{q\rightarrow 1} \frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}}$

Given the Euler function $\phi(q)=\prod_{n = 1}^{\infty}(1-q^{n})$ which is a modular form where $q=\exp(2\pi i \tau)$, $|q|\lt1$ Then what is the limit $\lim_{q\rightarrow 1}\frac{\phi^5(q)_{\...
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how to derive the formula of Jacobi theta function which is below

What is the derivative of Jacobi theta function which is: $$\Theta_3(z;\tau)= \sum_{n=-\infty}^\infty \exp(i\pi\tau n^2)\exp(2ni z)$$ and find its zeros.
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Analytic continuation of Riemann Theta function to positive semidefinite matrices

The Riemann $\Theta$-function is defined as $$Θ(z|Ω)=\sum\limits_{q∈Z^N}e^{πiq⋅Ωq+2πiq⋅z},$$ where $\Omega$ has positive definite imaginary part to ensure convergence. In a particle physics ...
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How are the Jacobi Theta Functions analogous to the Exponential Function?

On the Wolfram MathWorld page on Jacobi Theta Functions, it says that the Theta Functions are elliptic analogues of the exponential function. Is this because they satisfy certain properties that the ...
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1answer
157 views

What's the connection between $\theta$ series and the number of integer solutions on a curve? Proof Verification

I just started learning about theta series and am now flexing my muscles. I hope that everything looks good. This is a request for a proof verification. I am asking for what the proper name in the ...
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1answer
181 views

How many integer pairs satisfy the ellipse $x^2+ay^2=r?$

How many integer pairs satisfy the ellipse $x^2+ay^2=r?$ What I have discovered thus far: This post is largely to document the thinking that I have already done... I know that this can be frowned ...
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Clarification of Landau's Mechanics, Chapter.6 & Whittaker, Analytical Dynamics Chapter.6 needed

I am currently working through Landau's mechanics book and I am struggling to get my head around a solution provided in Chapter.6 relating to the asymmetrical top. I fully comprehend how Landau ...
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1answer
120 views

How many integer solutions are there on an $n$ dimensional hypersphere of radius $\sqrt{r}$ centered at the origin?

Let $\phi(n,r)$ be the number of integer solutions of $\sum\limits_{i=1}^n x_i^2=r$. Then $\phi(2,r)=4\sum\limits_{d|r}\chi(d)$ where $\chi (x)=sin(\frac{\pi x}{2})=\cases{ 1\text{ when }x\cong 1 \...
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$\sum\limits_{\mathbb{d|n}}{f(d)}=\sum\limits_{\mathbb{d|n}}{g(d)}\implies f(n)=g(n)?$

Question: Is it true that if for functions $f,g$ which map naturals to naturals For all natural numbers n, we have $f(n)=g(n) \iff$ for all natural numbers n we have $\sum\limits_{\mathbb{d|...
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1answer
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Properties of $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$?

I'm looking for references concerning the properties of the function $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$ where $\theta_{1}(z|\tau)$ is a Jacobi theta function defined here. I am trying ...
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$\Theta$ function in terms of Weierstraß $\sigma$ function?

Let $\Theta$ function be the function associated to a lattice $\Lambda=\oplus_{i\leq 2}Z\lambda_i\subset C$ of $C$ with transformation property defined as $\lambda\in\Lambda, \Theta(z+\lambda)=\Theta(...
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1answer
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$\theta(z) = \sum_{n=-\infty}^{+\infty} e^{\frac{-n^2}{2}}e^{inz}$ How to show $\theta '(\frac{i}{2}) = \frac{-i}{2} \theta (\frac{i}{2})$?

$$\theta(z) = \sum_{n=-\infty}^{+\infty} e^{\frac{-n^2}{2}}e^{inz}$$ How to show $\theta '(\frac{i}{2}) = \frac{i}{2} \theta (\frac{i}{2})$?
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Hecke Operators and Eigenfunctions, Fourier coefficients

The problem statement, all variables and given/known data Consider the action of $T_2$ acting on $M_k(\Gamma_{0}(N)) $, and show that $\theta^4(n)+16F $ and $F(t)$ are both eigenfunctions. Functions ...
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How to show that the theta function is smooth?

I was reading on the theta function $$ \vartheta(x) = \sum_{n=-\infty}^\infty e^{-\pi n^2 x}\qquad x>0 $$ The author claims that $\vartheta$ is smooth. Is there an easy way to show that this is ...
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Product of multi-dimensional theta functions

I am interested into the following sum of products of multi-dimensional theta functions on a genus-$g$ Riemann surface: $$ \sum_{\ell} \vartheta\!\!\begin{bmatrix} a + \frac{\ell}{k} \\ b_1 \end{...
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1answer
40 views

Basic $\theta$-function identity proof

For the $\theta$-function $$\theta (z) = \sum_{n \in \mathbb{z}} q^{n^2}e^{2\pi inz},$$ for $q$ given by $e^{\pi i\tau}$ for some $\tau \in \mathbb{C}$ with $Im(\tau) > 0$, suppose we've proved ...
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Asymptotic equivalent of $\sum_{n\ge0} q^{n^2}{x^n}$ as $x\to+\infty$

Let $q\in\Bbb C^*$ with $|q|<1$, define $$f:x\mapsto\sum_{n\ge0} q^{n^2}{x^n}$$ I want to find an asymptotic equivalent of $f$ as $x\to+\infty$. I found that $$a\le|f(x)|\cdot\exp\left(\frac{\...
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1answer
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The even-index reciprocal Lucas constant and $\sum_{n=1}^\infty \frac1{x_1^{2n}+x_2^{2n}}$

The sum of reciprocals of even index Lucas numbers has a nice closed-form in terms of theta functions, $$\begin{aligned}S_e &= \sum_{n=1}^\infty \frac1{L_{2n}}\\ &= \sum_{n=1}^\infty \frac1{\...
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proving sigma = BigTheta (BigΘ)

I'm trying to solve a BigΘ problem, and could use a little guidance to make sure I'm on the right path. So my question is to show that $\sum_{i=1}^{n} i^{15} = Θ(n^{16})$ I know that for something ...
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Borel -summation for Riemann-Siegel theta function

Riemann-Siegel theta function is defined in terms of the Gamma function as: $\theta(t)=\arg(\Gamma(\frac{1+2it}{4}))-\frac{\log \pi}{2}t $ , it has asymptotic series which is not converge and it's ...
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2answers
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An interesting identity involving Jacobi $\theta_4$ and $\zeta(2)$

A recent question mentioned an integral identity involving Dedekind $\eta$ function and a special value for the complete elliptic integral of the first kind. I refrained from providing a complete ...
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How to evaluate sums in the form $\sum_{k=-\infty}^\infty e^{-\pi n k^2}$

Online, one may find the values of the following sums: $$\sum_{k=-\infty}^\infty e^{-\pi k^2}=\frac{\pi^{1/4}}{\Gamma(3/4)}$$ $$\sum_{k=-\infty}^\infty e^{-2\pi k^2}=\frac{\pi^{1/4}(6+4\sqrt 2)^{1/4}}{...
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Evaluate $ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty $

This identity is taken from a physics paper [1] stated without proof, on page 43. $$ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty =...
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1answer
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Did Gauss know Jacobi's four squares theorem?

This is a question that i have already asked on HSM stackexchange, and i decided to ask it again here because it's more mathematical than historic (to make a conclusion in this question one needs more ...
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1answer
56 views

Linearly equivalent divisors induce same projective embedding

Consider the torus $T=\mathbb C^n/\Lambda$ for some lattice $\Lambda$. Say that two divisors $A,B$ on $T$ are linearly equivalent if their difference is the divisor of a meromorphic function on $T$. ...
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Can we extend the theta function $\theta(z)$ to p-adic numbers $\mathbb{Z}_p$?

Let $\theta(z) = \sum_{n \in \mathbb{Z}} q^{n^2}$ with $q = e^{2\pi i n z}$. Can we extend the theta function to $p$-adic arguments? Here's an example: $$ \theta( 1 + p^k) = \sum_{n \in \mathbb{Z}^...
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Has the idea of a 'hyperbolic Theta function' been studied?

Consider a uniform $\{m,n\}$ tiling of the hyperbolic plane, for convenience with one vertex at the origin (and also for convenience, normalize the edges to have unit hyperbolic length). Then there ...
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No weight $2$ newforms on $\Gamma_0(4)$?

How could it be there are no modular forms of weight $2$ and level $4$. In the lmfdb database? We can easily think of one: $\theta(z)^4$ with $\theta(z)=\sum_{n\in \mathbb{Z}} q^{n^2}$ . Where are ...
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1answer
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$ \sum_{n=1}^{\infty} \frac{ x^{n^2} (1 + x^n) - x^n}{1 -x^n} = 0.$ ??

While studying theta functions I noticed $$ \sum_{n=1}^{\infty} \frac{ x^{n^2} (1 + x^n) - x^n}{1 -x^n} = 0.$$ Why is that so ?? Is there a similar case with a term $x^{n^3}$ ??
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Theta notation for Strassen's multiplication

A student discovers a way to multiply 2×2 matrices using exactly 5 multiplications, instead of Strassen’s 7. What is the number M(n) of multiplications for the resulting algorithm to multiply n×n ...
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1answer
123 views

Representation of integers as quadratic forms with integer coefficients

While reading the book The sensual (quadratic) form by J.H. Conway I got curious in this question. Maybe it is trivial, but I don't know how to answer it. Let $f(x,y)=ax^2+hxy+by^2$, $g(x,y)=a'x^2+h'...
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1answer
266 views

Infinite series containing ceiling function

I am looking for a closed-form solution to this infinite (converging) series containing a ceiling function: \begin{equation}\sum_{k=1}^{\infty}\frac{1}{2^{2k+\lceil 2\sqrt{k} \rceil}} \approx 0....
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2answers
110 views

How can we show that $R=S?$

Let $$R=\prod_{n=1}^{\infty}\left({q^{2n-1}-1\over q^{2n-1}+1}\right)^2 \left({q^{2n}-1\over q^{2n}+1}\right)^2\tag1$$ and $$S=2\prod_{n=1}^{\infty}\left({q^{2n}+1\over q^{2n}-1}\right)^{2(-1)^{n+1}}...
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1answer
68 views

Do these summations satisfy modular properties

In this post ,I observed computationally that the mock theta functions of order $3$,found in this wikipedia article $f(q)=\sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q;q)^2_{n}}$,$\phi(q)=\sum_{n=0}^{\infty}...
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1answer
213 views

Modular transformation in terms of generators

I was looking for a "general" modular transformation for the first kind of Jacobi theta function, $\theta_1(u,\tau)$. I do know how this theta function transforms under the two generators of modular ...
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1answer
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conjectured relations of mock theta functions of order $3$

Given the following mock theta functions of order $3$,found in this wikipedia article $f(q)=\sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q;q)^2_{n}}$,$\phi(q)=\sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q^2;q^2)_{n}...
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133 views

A more general closed-form of an integral involving a square power of $\theta_4$ - function

$\textbf{Problem statement}$. Inspired by the computations at this nospoon we introduce the following integral: $$\int_{0}^{\infty }\frac{~\theta _{4}^{2}\left( \exp \left( -\pi \,y\,\beta \right) \...
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79 views

Reciprocal sums

How can I prove the following? 1)$$\sum_{k=0}^{\infty}\frac{1}{F_{2k+1}}=\frac{\sqrt{5}}{4}\theta_{2}^{2}(0,\frac{3-\sqrt{5}}{2}),$$ 2)$$\sum_{k=0}^{\infty}\frac{1}{F_{k}^{2}}=\frac{5}{24}(\theta_{2}^{...
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1answer
185 views

Applications of Jacobi theta functions [closed]

I've heard that the Jacobi theta function has wide applications, for instance in physics and proving elementary facts about the Fibonnaci sequence. I'd be interested in knowing some of the most ...
4
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0answers
59 views

Calculating $\Theta$ series of $E_8$ Lattice

I'm trying to calculate the $\Theta$ series of $E_8$ lattice, using the following Gram matrix (the Cartan Matrix of $E_8$): $$\left(\begin{matrix} 2 & 0 & 0 & -1 & 0 & 0 & 0 &...
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0answers
49 views

Discrete theta functions and periodicity

I'm doing quantum mechanics and I have an eigenfunction which is a theta function. I then discretised it, since I want see if I can find the eigenvalues for the discrete case by finding the ...