Asymptotic expansion of Legendre functions at $x=-1$

Question

Let $P_{\nu}(z),\,-1<z\leq1,\,\nu\in\mathbb{C}$ Legendre functions of the first kind defined as $$P_{\nu}(z)=\,_{2}F_{1}\left(-\nu,\nu+1;1;\frac{1-z}{2}\right).$$ I found this formula on Wolfram Alpha $$P_{\nu}\left(z\right)=\frac{\sin\left(\pi\nu\right)}{\pi}\log\left(\frac{z+1}{2}\right)\sum_{k=0}^{\infty}\frac{\left(-\nu\right)_{k}\left(\nu+1\right)_{k}}{k!^{2}}\left(\frac{z+1}{2}\right)^{k}$$ $$+\frac{\sin\left(\pi\nu\right)}{\pi}\sum_{k=0}^{\infty}\frac{\left(-\nu\right)_{k}\left(\nu+1\right)_{k}\left(2\psi\left(k+1\right)-\psi\left(k+\nu+1\right)-\psi\left(k-\nu\right)\right)}{k!^{2}}\left(\frac{z+1}{2}\right)^{k}$$ where $\psi(k)$ is the digamma function and where $\nu$ is not an integer and $\left|\frac{z+1}{2}\right|<1$. I read some manuals, like Olver “Asymptotic and special functions” and others but I'm not able to find a reference for such equation and I do not understand how to derive it. I think there is some link with $Q_{\nu}(z)$ but it is not obvious for me. Can someone give me some references for such identity or a proof of it?