Tagged Questions
1
vote
1answer
54 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
164 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
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 + ...
6
votes
1answer
537 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
$$\theta(z | r) = ...
3
votes
0answers
96 views
Relationship between Dixonian elliptic functions and Borwein cubic theta functions
In this paper, it says that the three Borwein cubic theta functions obey the identity $a(q)^{3}=b(q)^{3}+c(q)^{3}$, which is strongly reminiscent of the identity that Dixonian elliptic functions obey ...
10
votes
0answers
324 views
Hermite's solution of the general quintic in terms of theta functions
Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
5
votes
1answer
322 views
Where are this kind of series used, $\vartheta_{4}(0,e^{\alpha \cdot z})$?
In my recent explorations I stumbled upon the following series
$$
\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 ...
