There exist the linear identities for the 2f1 hypergeometric function where z is either -1, 1, or 1/2 using the quadratic transdormations it is easy to derive new identities in terms of gamma functions for z = -1/8,8/9 and 1/9

I have seen some identities with z=1/4 and z=-1/3 but there is no obvious transformation from the linear to the quadratic that yield these. So how do you derive them using the existing linear, quadratic transformation?

this problem i assure you is extremely difficult