In my project, I am dealing with the Gauss hypergeometric function, ${}_{2}F_{1}$, which enters my calculations through the following function:
$$\Psi(x) = x^{2-\gamma} \left[\frac{{}_{2}F_{1}\left(\frac{3-\gamma}{\alpha}, \frac{\beta-\gamma}{\alpha}; \frac{3-\gamma}{\alpha}+1; -x^\alpha\right)}{3-\gamma} - \frac{{}_{2}F_{1}\left(\frac{2-\gamma}{\alpha}, \frac{\beta-\gamma}{\alpha}; \frac{2-\gamma}{\alpha}+1; -x^\alpha\right)}{2-\gamma}\right]$$
Since I need to integrate numerically a product of first and second derivatives of $\Psi$, I was hoping there existed a way to simplify the expression above, as the integration quickly becomes very computationally expensive.
I have searched thoroughly the DLMF and reference therein, and I attempted several variable transformations on each individual ${}_{2}F_{1}$ term, in the hope of casting them into a format for which a simple analytical expression exist. However, I'm afraid that it cannot be achieved with the transformation reported in https://dlmf.nist.gov/15.8
I have also inserted the expression of $\Psi$ into Mathematica and let it simplify, but it simply returns the input.
Does a transformation exist which could scale all parameters by the same quantity , for instance one that could remove the $\frac{1}{\alpha}$ term?
Alternatively, can someone suggest a way to simplify $\Psi$ or cast it into another (possibly special) function which would allow to group the $_{2}F_{1}$ terms?
Thank you