For questions about $\theta$ functions.
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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 $S_r(n)$ ...
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:
$$\sum_{n=1}^\infty e^{- \pi n^2 x} < ...
10
votes
0answers
310 views
Hermite's solution of the general quintic in terms of theta functions
Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
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0answers
160 views
Convexity of $\theta(q)$
Define Jacobi's (fourth) theta function with argument zero and nome $q$:
$$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$
plot of the function via Wolfram|Alpha
plot of the function via Sage
I ...
5
votes
0answers
231 views
Extension of the Jacobi triple product identity
The Jacobi triple product identity is:
$$\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} $$
I would like to extend the idea for ...
5
votes
0answers
169 views
How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\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 ...
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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 ...
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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 + ...
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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 ...
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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 ...
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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 ...
