I was working on a problem involving intersection of hyperspheres in $\mathbb{R}^d$ and finding approximate volume of the intersection of two hyperspheres. There exists an explicit formula in terms of regularized incomplete Beta function. However, I was interested in bounding the volumes in forms of simple functions like polynomials and exponentials. Therefore, I am looking for lower bounds for regularized incomplete Beta function. In fact, to be specific, only of the form $\displaystyle I_x \left(\frac{n}{2}, \frac{1}{2} \right)$.
I tried all the basic stuff by approximating one of the terms and integrating the other. All such techniques resulted in bounds that were locally good. However, I am more interested in finding bounds that approximate the function well enough throughout the domain $[0,1]$. Another option is to possibly use Taylor Series, but I would want to avoid having the final expression in the form of a summation.
Any leads or references would be highly appreciated! Thanks a lot!
Since you do not want Taylor series, we can make them Padé approximants built around $x=0$ writing them for example $$P_m=\frac{x^{\frac n2}}{B\left(\frac{n}{2},\frac{1}{2}\right) }\frac{ \frac 2n +\sum_{i=1}^m a_i^{(m)} x^i}{1+\sum_{i=1}^m b_i^{(m)} x^i }$$ which seem to be systematic underestimates of $\displaystyle I_x \left(\frac{n}{2}, \frac{1}{2} \right)$ and good approximations except close to $x=1$. Remember that $P_m$ is at least $O(x^{2m+1})$.
For the simplest one, $$a_1^{(1)}=-\frac{n^2+4 n+12}{2 n (n+2) (n+4)}\qquad \text{and} \qquad b_1^{(1)}=-\frac{3 (n+2)}{4 (n+4)}$$ I also built $P_2$ but the coefficients start to be quite messy.