For questions about $\theta$ functions.
5
votes
0answers
229 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 ...
3
votes
1answer
73 views
series involving $\log \left(\tanh\frac{\pi k}{2} \right)$
I found an interesting series
$$\sum_{k=1}^\infty \log \left(\tanh \frac{\pi k}{2} \right)=\log(\vartheta_4(e^{-\pi}))=\log \left(\frac{\pi^{\frac{1}{4}}}{2^{\frac{1}{4}}\Gamma \left( ...
1
vote
1answer
52 views
Coefficients of powers of the theta function
Let $q=\exp(2 \pi i z)$ and $$\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$$
where $S_r(n)$ ...
9
votes
0answers
160 views
Convexity of $\theta(q)$
Define Jacobi's (fourth) theta function with argument zero and nome $q$:
$$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$
plot of the function via Wolfram|Alpha
plot of the function via Sage
I ...
1
vote
1answer
42 views
Jacobi Theta function inequality
I am trying to show that
$$\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
63 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 ...
0
votes
0answers
43 views
Interpretation of functional equation of dedekind eta function
It's well known that the Dedekind eta function defined by $\eta(z) = \displaystyle e^{\frac{\pi i z}{12}} \prod_{n=1}^{\infty} (1 - e^{2 \pi i n z}) $ converges for $z$ in the upper half plane to a ...
1
vote
1answer
104 views
Big Theta Expression Question
I need help - I have to express this function:
$f(n)=15n \log n + 12n + 9 \log n +25$ in terms of big theta notation. I believe that it is $\Theta(n \log n)$, but I have to prove it mathematically. ...
1
vote
2answers
101 views
Question about theta of $T(n)=4T(n/5)+n$
I have this recurrence relation $T(n)=4T(\frac{n}{5})+n$ with the base case $T(x)=1$ when $x\leq5$. I want to solve it and find it's $\theta$. I think i have solved it correctly but I can't get the ...
3
votes
1answer
180 views
Calculating a summation of a $\theta$ function
Let $ \theta_z(t) = \sum \limits_{m,n\in\mathbb{z}}e^{-\pi Q_z(m,n)t}$ where $Q_z(m,n)=y^{-1}|mz+n|^2$.
I need to prove that $\theta_z(t)=t^{-1}\theta_z(t^{-1})$.
Now, looking that up I know that ...
0
votes
1answer
89 views
Is this why this summation is equivalent to this Theta notation?
So I'm not sure if I misunderstood the lesson or not.
$$T(n) =\sum_{j=2}^{n}\Theta(j) = \Theta(n^2) $$
Are these equivalent because:
$$ \sum_{j=2}^{n}\Theta(j) = \frac{n(n-1)}2 - \frac{1(1 - 1)}{2} = ...
0
votes
1answer
80 views
Showing a recurrence is $\Theta$(n)
Specifically how do you go about showing that
$$
2T(n/2)+1 =\Theta(n)
$$
Not looking for an answer, as much as the process? I'm studying for a test and this is one of the review problems. Thanks in ...
0
votes
0answers
58 views
Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)
I need to evaluate this series for arbitrary $\beta > 0$:
$ Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $
Is it related to a known transcendental function?
From the research I did, it ...
11
votes
2answers
215 views
Combinatorial interpretation of this identity of Gauss?
Gauss came up with some bizarre identities, namely
$$
\sum_{n\in\mathbb{Z}}(-1)^nq^{n^2}=\prod_{k\geq 1}\frac{1-q^k}{1+q^k}.
$$
How can this be interpreted combinatorially? It strikes me as being ...
6
votes
1answer
527 views
On the growth of the Jacobi theta function
So, I ran into this exercise from Stein & Shakarchi. CA, Chapter 5:
Show that if $\tau$ is fixed with positive imaginary part, then the Jacobi theta function
$$\theta(z | r) = ...
1
vote
0answers
95 views
A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+… = 0$
Given the Dedekind eta function $\eta(\tau)$. Define,
$y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$
Prove the multi-grade identity [1],
$y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + ...
5
votes
0answers
169 views
How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
1
vote
0answers
130 views
What is meant by a set of generic points on a compact Riemann surface
Let $X$ be a compact connected Riemann surface of genus $g \geq 1$.
I'm studying a theorem of Faltings which looks as follows.
Let $P_1,\ldots, P_g$ be generic points on $X$. Then we have some ...
5
votes
1answer
322 views
Where are this kind of series used, $\vartheta_{4}(0,e^{\alpha \cdot z})$?
In my recent explorations I stumbled upon the following series
$$
\vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z ...
4
votes
1answer
76 views
Reference request: theta functions for lattices which are not unimodular or even
In A Course in Arithmetic, Serre works out some of the theory of theta functions for even, self-dual lattices, e.g. such theta functions are modular forms of weight $n/2$, where $n$ is the dimension ...
3
votes
0answers
94 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 ...
10
votes
0answers
309 views
Hermite's solution of the general quintic in terms of theta functions
Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
