Efficient evaluation of hypergeometric functions

Question

Does anyone have experience with algorithms for evaluating hypergeometric functions? I would be interested in general references, but I'll describe my particular problem in case someone has dealt with it.

My specific problem is evaluating a function of the form 3F2(a, b, 1; c, d; 1) where a, b, c, and d are all positive reals and c+d > a+b+1. There are many special cases that have a closed-form formula, but as far as I know there are no such formulas in general. The power series centered at zero converges at 1, but very slowly; the ratio of consecutive coefficients goes to 1 in the limit. Maybe something like Aitken acceleration would help?

Answer 1

Is it correct that you want to sum a series where you know the ratio of successive terms and it is a rational function?

I think Gosper's algorithm and the rest of the tools for proving hypergeometric identities (and finding them) do exactly this, right? (See Wilf and Zielberger's A=B book online.)

Answer 2

I tested Aitken acceleration and it does not seem to help for this problem (nor does Richardson extrapolation). This probably means Pade approximation doesn't work either. I might have done something wrong though, so by all means try it for yourself.

I can think of two approaches.

One is to evaluate the series at some point such as z = 0.5 where convergence is rapid to get an initial value and then step forward to z = 1 by plugging the hypergeometric differential equation into an ODE solver. I don't know how well this works in practice; it might not, due to z = 1 being a singularity (if I recall correctly).

The second is to use the definition of 3F2 in terms of the Meijer G-function. The contour integral defining the Meijer G-function can be evaluated numerically by applying Gaussian or doubly-exponential quadrature to segments of the contour. This is not terribly efficient, but it should work, and it should scale to relatively high precision.