Board index Special Functions A bizarre elliptic identity

A bizarre elliptic identity

Post your questions related to Special Functions here.

Moderator: Shobhit

A bizarre elliptic identity

Wed Dec 18, 2013 10:48 am

Posts: 38
Location: India, West Bengal
$\displaystyle \frac{\sigma(nz)}{\sigma^{n^2}(z)} = \frac{(-1)^{n-1}}{\left(1!2!3!\cdots(n-1)!\right)^2}\begin{vmatrix} \wp'(z) & \wp''(z) & \cdots & \wp^{(n-1)}(z)\\ \wp''(z) & \wp'''(z) & \cdots & \wp^{(n)}(z)\\ . & . & \ldots & .\\ . & . & \ldots & .\\ \wp^{(n-1)}(z) & \wp^{(n)}(z) & \cdots & \wp^{(2n-3)}(z) \end{vmatrix}$

Do you guys have any ideas? Our king of elliptic functions, perhaps?

Re: A bizarre elliptic identity

Tue Jan 28, 2014 10:32 am

Posts: 8
This equation is known as the "Frobenius–Stickelberger Addition Formula". The following papers might be of interest to you:

1. http://www.ccn.yamanashi.ac.jp/~yonishi/research/pub/fs.pdf

2. http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=34B0325E9BBD94D474181C9EFEF7CA9A?doi=10.1.1.298.6730&rep=rep1&type=pdf
Last edited by Ramanujan on Wed Jan 29, 2014 1:49 am, edited 1 time in total.

Re: A bizarre elliptic identity

Tue Jan 28, 2014 4:48 pm

Posts: 38
Location: India, West Bengal
I see. I found this one on my way through my research on quintics, and I was not aware that this was done by Frobenius and Stickelberger.

But F-S formula seems to be a generalization of what I actually wanted, which I got a long ago from Schwarz-Weierstrass.

Return to Special Functions