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A bizarre elliptic identity

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Post 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? :D

Post 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.

Post 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.


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