I was doing some norm estimates for the spherical mean value operators (motivated from this paper). In the process, I have in hand the following integral:
$$\int\limits_{-1}^{1} \left( 1 - t \right)^{\tilde{p} \left( \frac{\rho + 3}{2r} - 1 - B + \frac{2 \alpha + 2}{q} \right) - 1} \left( 1 + t \right)^{\tilde{p} \left( \frac{-3\rho - 1}{2} +2 \alpha + 1 + B - \frac{2\alpha + 2}{q} \right)} {}_2F_1 \left( r\alpha - \frac{\rho - 1}{2}, r \left( \alpha - \frac{1}{2} \right) + 1; r \left( 2 \alpha - 1 \right) + 2; 1 - t^2 \right)^{\frac{\tilde{p}}{r}} \mathrm{d}t$$
At first, I was hoping to simplify the argument of the Hypergeometric function. To that end, I searched the NIST handbook of mathematical functions, and the Wolfram website. However, I could not find any transformation of variables that simplifies the job!
Next, it seems that this question could help. However, getting the arguments of this question in terms of what I have seems to be difficult. any hints or insights into calculating this integral would be appreciated!
PS: Although we can estimate this integral by using the fact that $1 + t$ and $1 - t$ are both bounded above by $2$, I believe that it will become a crude estimate and will mess up with calculations I want to do further. Therefore, as far as possible, I wish to evaluate this integral explicitly.