I need a numerically stable way to compute the following ratio:
All the parameters are real positive numbers, with $0 < a,b,c$ and $0 < x < 1$.
Right now I am using GSL's implementation of the hypergeometric function, but I keep getting under/over-flows.
Is there a simplification that I can use? Or an aproximation scheme that remains accurate for a wide range of the parameters (say $0 < a,b,c < 1000$)?