Relation between full elliptic integrals of the first and third kind I am working on a calculation involving the Ronkin function of a hyperplane in 3-space.
I get a horrible matrix with full elliptic integrals as entries. A priori I know that the matrix is symmetrical and that give me a relation between full elliptic integrals of the first and third kind.
I can not find transformations in the literature that explain the relation and I think I need one in order to simplify my matrix.
The relation
With the notation
$\operatorname{K}(k) = \int_0^{\frac{\pi}{2}}\frac{d\varphi}{\sqrt{1-k^2\sin^2\varphi}},$
$\qquad$ $\Pi(\alpha^2,k)=\int_0^{\frac{\pi}{2}}\frac{d\varphi}{(1-\alpha^2\sin^2\varphi)\sqrt{1-k^2\sin^2\varphi}}$
$k^2 = \frac{(1+a+b-c)(1+a-b+c)(1-a+b+c)(-1+a+b+c)}{16abc},\quad a,b,c > 0$
the following is true:
$2\frac{(1+a+b-c)(1-a-b+c)(a-b)}{(a-c)(b-c)}\operatorname{K}(k)+$
$(1-a-b+c)(1+a-b-c)\Pi\left( \frac{(1+a-b+c)(-1+a+b+c)}{4ac},k\right) +$
$\frac{(a+c)(1+b)(1-a-b+c)(-1-a+b+c)}{(a-c)(-1+b)}\Pi\left( \frac{(1+a-b+c)(-1+a+b+c)(a-c)^2}{4ac(1-b)^2},k\right)+$
$(1-a-b+c)(-1+a+b+c)\Pi\left( \frac{(1-a+b+c)(-1+a+b+c)}{4ac},k\right)+$
$\frac{(1+a)(b+c)(-1-a+b+c)(-1+a+b-c)}{(1-a)(c-b)}\Pi\left( \frac{(1-a+b+c)(-1+a+b+c)(b-c)^2}{4ac(1-a)^2},k\right)$
$==0$.
Is there some addition formula or transformation between elliptic integrals of the first and third kind that will explain this? 

Have you tried MGfun from Frédéric Chyzak?  It is a (Maple) package to deal with computations in multivariate Ore Algebras.  The main application of this is exactly the kind of problem you pose: finding relations between (multivariate) holonomic functions.  It is a generalization of WZ theory to the multi-variate case.
