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][1] 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][2]. $G$ 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. [1]: http://www.cs.cmu.edu/~adamchik/articles/sums/Csum.pdf [2]: http://mathworld.wolfram.com/Dilogarithm.html