I was looking at the Complete Elliptic Integral of the First Kind a few days ago:

$K(k) = \int_{0}^{1} \frac{dx}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}$

And it occurred to me that the denominator kind of resembles the first two terms of the product defined by a $q$-Pochhammer symbol. The natural thing to do would then seem to be:

$V(k)= \int_{0}^{1} \frac{dt}{\sqrt{(t^{2} ; k^{2})_{\infty}}}$

I'd like to be able to extend results to ordinary elliptic integrals to this beast.

Preliminary experiments with elliptic integral singular values are promising:

$k_{3} = (\sqrt{6}-\sqrt{2})/4$:

>>> quad(lambda x: 1/sqrt(qp(x*x,k3*k3)), [0,1])\\

mpf('1.60008504344303782384841188109928')\\

>>> sqrt(pi) * gamma(1/6) / (2* 3**(3/4.0) * gamma(2/3.0))\\

mpf('1.59814200211254025211306432770042')\\

I'd like to be able to evaluate $V(k)$ in terms of known special functions. But that square root complicates

things, and I'm not sure what to do next.

Owen