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$\begingroup$

I'm looking for any "closed form" for the coefficient of the $\ell$-th power of $K(x)$, the complete elliptic integral of the first kind.

Thanks.

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    Does (2) from [here](http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html) help you? [Wiki](http://en.wikipedia.org/wiki/Elliptic_integral) also has something to tell :$K(k) = \frac{\pi}{2} \sum_{n=0}^\infty \left[\frac{(2n)!}{2^{2 n} (n!)^2}\right]^2 k^{2n} = \frac{\pi}{2} \sum_{n=0}^\infty [P_{2 n}(0)]^2 k^{2n}$.2012-04-23
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    Thanks for answer. I have this expression and I look for the l-th power. I have found a connexion with the Dedekind eta function for which l-th powers are well studied...let see...;o)). Thanks again.2012-04-23

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