A calculation with Bessel functions

Question

I'm reading "Introduction to Quantum Effects in Gravity - Mukhanov, Winitzki" and despite treating purely physical subjects, I'm stuck with a mathematical calculation that I suspect involve some properties of Bessel or Gamma functions that I don't know. I have this expression $$ f \doteq \sqrt{a|x|} ( A J_\alpha (a|x|) + B Y_\alpha (a|x|) ) $$ where $a$ and $\alpha$ are some real positive values, and a constraint $$ \frac{\text{d}f}{\text{d}x} f^\ast - f \frac{\text{d}f^\ast}{\text{d}x}=2\mathrm{i} $$ In the book (7.3.3 pag. 96) is stated that this brings to $$ A B^\ast - A^\ast B = \frac{\mathrm{i}\pi}{a} $$ but I totally don't see how. I tried doing an explicit calculation, but I'm stuck in trying to demonstrate that $$ \frac{\text{sgn}\,x}{|x|} \sum_{n,m=0}^{+\infty} \frac{ (-1)^{n+m} (\alpha+n-m) }{ n!m! \Gamma(n+\alpha+1) \Gamma(m-\alpha+1) } \left( \frac{a|x|}{2} \right)^{2(n+m)} = -\pi\sin{\pi\alpha} $$ Also the explicit way doesn't seem very pleasant to me, but trying to start from differential Bessel equation and manipulating a bit doesn't get me to any result. What is the way to do it?