Acceleration of generalized hypergeometric functions through precise remainder asymptotics

Details Acceleration of generalized hypergeometric functions through precise remainder asymptotics Acceleration of generalized hypergeometric functions through precise remainder asymptotics

2011-02-15

arxiv.org/abs/1102.3003v2

Num. Alg. 59 (2012) 447-485

Mathematics

Numerical Analysis

36 pages, 6 figures, LaTeX2e. Fixed sign error in Eq. (2.28), added several references, added comparison to other methods, and

Scientific paper

10.1007/s11075-011-9499-9

We express the asymptotics of the remainders of the partial sums {s_n} of the generalized hypergeometric function q+1_F_q through an inverse power series z^n n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k} may be recursively computed to any desired order from the hypergeometric parameters and argument. From this we derive a new series acceleration technique that can be applied to any such function, even with complex parameters and at the branch point z=1. For moderate parameters (up to approximately ten) a C implementation at fixed precision is very effective at computing these functions; for larger parameters an implementation in higher than machine precision would be needed. Even for larger parameters, however, our C implementation is able to correctly determine whether or not it has converged; and when it converges, its estimate of its error is accurate.

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