I need a numerically stable way to compute the following ratio:$$\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$$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 $0 < a,b,c < 1000$)?
Compute hypergeometric function ratio: $\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$, for large positive $a,b,c$?
floating point;special functions