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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 help you? Wiki also has something to tell :$K(k) = \frac{\pi}{2} \sum_{n=0}^\infty \left[\frac{(2n)!}{2^{2 n} (n!)^2}\right]^2 k^{2n} = \frac{\pi}{2} \sum_{n=0}^\infty [P_{2 n}(0)]^2 k^{2n}$. –  draks ... Apr 23 '12 at 13:53
    
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. –  Gianfranco OLDANI Apr 23 '12 at 21:37
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2 Answers

Since the user seems to have not come back, I'll just leave this around:

Letting

$$K(m)=\int_0^\frac{\pi}{2}\frac{\mathrm du}{\sqrt{1-m\sin^2 u}}$$

be the complete elliptic integral of the first kind with parameter $m$, we have the corresponding hypergeometric series expansion

$$K(m)=\frac{\pi}{2}\sum_{j=0}^\infty \binom{2k}{k}^2\left(\frac{m}{16}\right)^k$$

As always, if one wants the coefficients of $\left(\dfrac2{\pi}K(m)\right)^\ell$, the usual recourse is the Faà di Bruno formula. One especially convenient form makes use of the partial Bell polynomials:

$$\left(\frac2{\pi}K(m)\right)^\ell=1+\sum_{j=1}^\infty \mathcal K_j m^j$$

where

$$\mathcal K_j=\frac1{j!}\sum_{r=1}^j r!\binom{\ell}{r} B_{j,r}\left(\frac14,\frac9{32},\cdots,\frac{(j-r+1)!}{16^{j-r+1}}\binom{2(j-r+1)}{j-r+1}^2\right)$$

There may be simpler expressions for $\mathcal K_j$, but I don't know them.


The convenience of the Bell polynomial formulation of the Faà di Bruno formula lies in the fact that computing environments like Mathematica explicitly support the partial Bell polynomials. As a demonstration:

With[{l = 5}, CoefficientList[Series[(2EllipticK[m]/Pi)^l,{m, 0, 6}],m]]
{1, 5/4, 85/64, 345/256, 22005/16384, 87009/65536, 1370745/1048576}

With[{l = 5}, Prepend[Table[
     Sum[k! Binomial[l, k] BellY[n, k,
              Table[j! Binomial[2 j, j]^2/16^j, {j, 1, n - k + 1}]],
         {k, 1, n}]/n!, {n, 1, 6}], 1]]
{1, 5/4, 85/64, 345/256, 22005/16384, 87009/65536, 1370745/1048576}
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According to Maple 16:

FunctionAdvisor(sum_form,EllipticK(x));

$$[{\it EllipticK} \left( x \right) =\sum _{{\it \_n1}=0}^{\infty }1/2\, {\frac {\pi \, \left( {\it pochhammer} \left( 1/2,{\it \_n1} \right) \right) ^{2}{x}^{2\,{\it \_n1}}}{ \left( {\it \_n1}! \right) ^{2}}},{ \it And} \left( \left| x \right| <1,x\neq 0 \right) ] $$ $$[{\it EllipticK} \left( x \right) =\sum _{{\it \_n1}=0}^{\infty }{ \frac { \left( {\it pochhammer} \left( 1/2,{\it \_n1} \right) \right) ^{2} \left( 1/2\,\ln \left( -{x}^{2} \right) +\Psi \left( 1+ {\it \_n1} \right) -\Psi \left( 1/2-{\it \_n1} \right) \right) }{{x}^ {2\,{\it \_n1}} \left( {\it \_n1}! \right) ^{2}}}{\frac {1}{\sqrt {-{x }^{2}}}},{\it And} \left( 1\leq \left| x \right| \right) ] $$

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