This question originates from [this recent question](https://math.stackexchange.com/questions/2503975/ramanujans-series-1-sum-n-1-infty8n1-left-frac1-cdot-5-cdots-4n-3) of Paramanand Singh about a series computed by Ramanujan, probably related to elliptic integrals and Legendre functions.

> 
> Is there a closed form for
>  $$ {}\_4 F\_3\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};1,1,1;1\right)=\sum\_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4=\frac{8}{\pi^3}\int\_{0}^{\frac{1}{2}}\frac{K(m)^2}{\sqrt{m(1-m)}}\,dm $$
>  ?
> 
> 
> 

Many proofs of $\sum\_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3=\frac{\pi}{\Gamma\left(\frac{3}{4}\right)^4}$ are well-known, for instance through Clausen formula or Fourier-Legendre series expansions (pages 27-28 [here](https://arxiv.org/pdf/1710.03221)). Such methods do not seem to apply smoothly for computing a closed form for the RHS, neither Parseval's identity applied to
$$ \sum\_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2 e^{ni\theta} = \frac{2}{\pi}\,K(e^{i\theta})$$
where $e^{i\theta}$ is regarded as the elliptic modulus. Suggestions are welcome.