I know it's not best idea now, but I just tried to implement Taylor series, because I don't have the book right now. If it is interesting, please look at https://github.com/AntonBikineev/math/blob/hypergeometric/include/ boost/math/special_functions/detail/hypergeometric_pfq_series.hpp
Also I have some ideas about reducing division operators in sum and thereby reducing rounding errors.
Nice. I guess the complicated part of implementing these functions is figuring out how to divide up the input domain: 1) When these series can safely be used. 2) When the result is a domain error. 3) When the result is necessarily infinity (series diverges to +INF). 4) When you can use recurrence relations and other identities to move closer to (1), and if those relations are stable. If the recursion is unstable then can something like Millers algorithm be used. Cheers, John.