Something is strange with $\sf LegendreQ$. Let $x>1$. I wonder why $\sf LegendreQ[\frac12,x]$ is complex-valued, and the following two codes do not give the same results:
(whose identity is given by Equation 8.820.2 of Gradshteyn & Ryzhik, 7th Ed.)
Thank you for your interest and time.