EDIT: Context for this investigation can be found in one of my other MO posts, "Pythagorean Theorem for Right-Corner Hyperbolic Simplices?"
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I'm investigating a function that has led me to this series:
$$T(r) = \sum_{k=0}^{\infty}{2k \choose k}\frac{1}{(k+r)^2 16^k}={}_3F_2(\;\;\frac{1}{2},\;r,\;r\;;\;1+r,\;1+r\;;\;\frac{1}{4}\;\;)\frac{1}{r^2}$$
This paper ("A certain series associated with Catalan's constant") by Victor Adamchik spotlights a tantalizingly similar series ...
$$S(r)=\sum_{k=0}^{\infty}{2k\choose k}^2 \frac{1}{(k+r) \;\; 16^k}={}_3F_2(\;\;\frac{1}{2},\;\frac{1}{2},\;r\;;\;1,\;1+r\;;\;1\;\;)\frac{1}{r}$$
... where the square is on the "wrong" factor.
I'm not very familiar with hypergeometric series (yet). Is there an identity that relates $S$ and $T$?
Edit: I've added the "${}_3F_2$" representations of $S$ and $T$.
Edit2: I have a particular interest in the case $r=1/2$:
$$T(1/2) = 4 \cdot {}_3F_2(\;\;\frac{1}{2},\;\frac{1}{2},\;\frac{1}{2}\;;\;\frac{3}{2},\;\frac{3}{2}\;;\;\frac{1}{4}\;\;) = 4 \cdot \Im( Li_2( \exp( \frac{i \pi}{3} ))) = 4 \;\; \sum_{k=1}^{\infty}\frac{\sin(\frac{\pi}{3}k)}{k^2}$$
$$S(1/2)= 2\cdot {}_3F_2(\;\;\frac{1}{2},\;\frac{1}{2},\;\frac{1}{2}\;;\;1,\;\frac{3}{2}\;;\;1\;\;) = \frac{8G}{\pi}$$
where $G$ is the Catalan constant
$$G = \sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^2} = {}_3 F_2(\;\; \frac{1}{2},\;\frac{1}{2},\;1 \;;\;\frac{3}{2},\;\frac{3}{2}\;;\;-1\;\;)$$
and $Li_2$ is the dilogarithm.
$G$ arises in my analysis of $T$, but the connection isn't clear (to me).
Edit3: Thanks to guidance from the comments, I've expressed $T(1/2)$ in terms of the dilogarithm function. I've also given the "${}_3F_2$" representation of $G$. The connection between $S(1/2)$ (or $G$) and $T(1/2)$ still eludes me, but I've only begun searching through the various caches of identities that have been recommended.
