I'm having trouble finding the Big-Theta asymptotic limit of expression involving incomplete beta function.
$S(k,k)=k\big(\frac{1}{\delta_1}-\frac{1}{\delta_2}\big)+2k^2\dbinom{2k}{k}\bigg[\frac{I_x(k,k+1)}{\delta_2}-\frac{I_x(k+1,k)}{\delta_1}\bigg]$
where
$I_x(k,k+1)$ and $I_x(k+1,k)$ are normalized incomplete beta functions with $x=\frac{\delta_1}{\delta_1+\delta_2}$
Remark: $\delta_1,\delta_2>0$ are constant and $k\in \mathbb Z_{\geq1}$