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Questions tagged [theta-functions]

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

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Properties of the Theta function

Define the $\vartheta :\mathbb{R}^+ \to \mathbb{R}$ by $$\vartheta (s) = \sum_{m=-\infty}^{\infty} e^{-\pi m^2s}$$ Is this a smooth ($C^{\infty}$) function? I would like to think so, but I'm not ...
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Addition formula for elliptic integral of second kind

Let $k\in(0,1)$ and the incomplete elliptic integral integral $E(u, k) $ be defined by $$E(u, k) =\int_{0}^{u}\operatorname {dn} ^2(t,k)\,dt\tag{1}$$ where $\operatorname {dn} (u, k) $ represents one ...
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Asymptotic behavior of $2\sum_{i=0}^n 4^i$

I know that the summation $$2\sum_{i=0}^n 4^i$$ exhibits $\theta(2^{n*ln(n)})$ asymptotic behavior. How do I prove this, however? I know that as $n\rightarrow \infty$ the value become ~ $2*(n+1)*4^...
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Characteristic functional equation of a Theta Function

Define the following as a "simple" theta function $$ \vartheta(q) = \sum_{n=0}^{\infty} q^{n^2} = 1 + q + q^4+q^9+ \ ...$$ Defined on the open unit circle on the complex plane. I'm trying to find ...
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How to prove that a division of $\vartheta(z:\tau)$ functions has no poles

How do I prove that $f(z) + f(iz)$ has no poles? where $$f(z)=\dfrac{\vartheta^2(0;i)\vartheta^2(z+ \frac{1}{2};i)}{\vartheta^2(\frac{1}{2};i)\vartheta^2(z;i)}$$ and $$\vartheta(z;\tau)=\sum\limits_{n=...
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Jacobi Triple Product using Euler's identities

I recently came across a simple proof for Jacobi's triple product (here), in the proof Andrews assumes two identities: $$ E_1 = \prod_{n=0}^{\infty}(1+x^nz) = \sum_{n=0}^{\infty}\frac{x^{n(n-1)/2}z^n}...
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Lattice associated to 4th Jacobi theta function?

For a lattice (specifically the dual lattice of a torus) there is associated a theta function $ \theta_{\Gamma}(w)=\sum_{\gamma\in\Gamma}w^{||\gamma||^2},\text{ where $w=e^{-4\pi^2t}$ and $t\in(0,\...
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1answer
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Theta function squared is a weight $1$ modular form

Let $$\vartheta(\tau) = \sum_{n\in\mathbb{Z}}e^{\pi in^2\tau}.$$ I know that $\vartheta$ satisfies the transfromation properties $$\vartheta(\tau + 2) = \vartheta(\tau), \quad \vartheta\left(-\frac{1}{...
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Asymptotics of Jacobi's third theta function.

For $z\in \mathbb{C}$ and $\tau \in i\mathbb{R}_{+}$ consider the function $$ \theta_3(z;\tau)=\sum_{n\in\mathbb{Z}}\exp\left(2\pi i n z +\pi i \tau n^2\right) $$ This function satisfies the well ...
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What is a closed form partial sum formula for the q-digamma function?

It is known that the partial sum of the digamma function can be expressed in closed form, but what of the q-digamma function? $$\sum_{x=1}^n\psi_q{\small(x)}\ =\ ?$$ $$q\in\mathbb N$$
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What does it mean when the exponent is beneath the base number?

I am familiar with the exponential function and the power function. Exponential Function: $y=3^x$ Power of Function: $y=x^3$ But what do the numbers under the theta in this image represent? $h_𝜃(...
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Infinite sum of 1/sin^2 and theta function

In studying some physical propagator, I came across the following sum $$ \sum_{n = -\infty}^{+\infty} \frac{ a^n }{ \sin^2(z + n \pi \tau) }\ . $$ Obviously, my question is how to evaluate this sum. ...
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Double sum of Lambert series: Partial sum in closed form desired!

We desire the things stated in the title for: $\sum_{k=2}^m \sum_{n=1}^{k-1} {q^n\over {1-q^n}}$ Some things I've looked into that may be of some help: The first sum is just a truncated (partially ...
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Cutting out a Riemann surface inside its Jacobian variety

After choosing a base point $P_{0}$ in a compact Riemann surface $X$ of genus $g$, the Abel-Jacobi map gives an embedding of $X$ into its Jacobian variety $Jac(X).$ This map can also be extended to ...
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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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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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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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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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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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202 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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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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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 $\lambda\in\Lambda, \Theta(z+\lambda)=\Theta(...
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$\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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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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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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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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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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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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$ \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 ...