Recently, I encountered the following series:
\begin{equation} \mathcal{I} = \sum_{q=1}^{\infty}\frac{\Gamma \left(q+\frac{1}{2}\right)^2 \left(2 q \psi ^{(0)}(q)-2 q \psi ^{(0)}\left(q+\frac{1}{2}\right)+1\right)}{\Gamma (q+1)^2}\, \ , \end{equation} where $\psi^{(0)}$ represents the digamma function. By using numerical methods, I have determined that this series should converge to the following value: \begin{equation} \mathcal{I} = -1+\pi-\frac{\pi^2}{4} \end{equation} This result appears promising. I am curious if anyone knows whether this series has been analytically summed, or if there are suggestions for summing it analytically. Your insights would be greatly appreciated.
Many thanks