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

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17
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1answer
387 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 ...
3
votes
0answers
38 views

Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{10}\left(e^{-\pi x}\right)}{1+x^2} dx $

Motivation 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 ...
9
votes
1answer
311 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 $k$ ...
1
vote
0answers
14 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. ...
2
votes
0answers
21 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
15 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( - ...
1
vote
0answers
46 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
53 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 ...
7
votes
1answer
291 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 ...
3
votes
0answers
28 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 + ...
3
votes
0answers
47 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, ...
0
votes
1answer
71 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: $|x| <1$
4
votes
1answer
140 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 ...
0
votes
1answer
72 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 ...
2
votes
1answer
38 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} ...
0
votes
1answer
80 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 ...
5
votes
0answers
162 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 ...
2
votes
0answers
78 views

Primitive of Weierstrass $\wp$

Consider a lattice $L=\mathbb{Z}+\mathbb{Z}\tau$. Take the function $\xi(z) = \frac{-1}{z} - \sum_{w \in L\backslash \{0\}} \Big ( \frac{1}{z-w} + \frac{1}{w} + \frac{z}{w^2} \Big )$. Obviously this ...
2
votes
1answer
50 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
0answers
64 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}} ...
6
votes
1answer
163 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( ...
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) = ...
4
votes
1answer
77 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
29 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
44 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
29 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 ...
5
votes
1answer
122 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 ...
1
vote
0answers
25 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) &= ...
2
votes
3answers
81 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
votes
0answers
44 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 ...
1
vote
0answers
25 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 ...
5
votes
2answers
170 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= ...
5
votes
2answers
314 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 ...
8
votes
2answers
150 views
7
votes
1answer
135 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 ...
1
vote
0answers
52 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 ...
8
votes
1answer
150 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 ...
2
votes
0answers
67 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} ...
4
votes
2answers
114 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, ...
3
votes
0answers
90 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 ...
1
vote
2answers
80 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}) \\ &= ...
2
votes
1answer
102 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, ...
4
votes
1answer
71 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 ...
4
votes
1answer
99 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 ...
1
vote
0answers
72 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
votes
0answers
41 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) = ...
2
votes
1answer
26 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 = ...
1
vote
0answers
15 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 ...
1
vote
0answers
38 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
47 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 $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}+F_{2k-1}}=\frac{(2k-1)\sqrt{5}}{2F_{2k-1}}$$ ...