Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in reasonable time, with all found digits provably correct)? I am particularly interested in this case: $$\mathcal{D}=\left.\partial_\alpha G_{2,3}^{2,1}\left(1\middle|\begin{array}c1,\alpha\\1,1,0\end{array}\right)\right|_{\alpha=1}$$

Solution 1:

Yes, it is possible in some cases. For example, $$\begin{align}\mathcal{D}&={_2F_2}\left(\begin{array}c1,1\\2,2\end{array}\middle|-1\right)\\&=\gamma-\operatorname{Ei}(-1),\end{align}$$ where ${_pF_q}$ is the generalized hypergeometric function, $\gamma$ is the Euler–Mascheroni constant, and $\operatorname{Ei}(z)$ is the exponential integral. In case you need a numeric value, $$\mathcal{D}\approx0.7965995992970531342836758655425240800732066293468318063837458...$$