Evaluating the limit of a function involving exponential and hypergeometric functions as it approaches infinity.

Question

I am trying to evaluate the limit of $$ e^{-BY} \left[ B \,e^{2Y} {}_2F_1 \left( 1,1-\frac{B}{2}; 2-\frac{B}{2}; -e^{-2Y} \right) -(B-2)\, {}_2F_1 \left( 1,-\frac{B}{2}; 1-\frac{B}{2}; -e^{-2Y} \right) \right] \, , $$ as $Y \to \infty$.

For $B=1$, it appears that the limit is $\pi$. I'm curious if a closed analytical form can be derived for arbitrary values of $B$. Any assistance on this matter would be greatly appreciated. Thank you!