Acceleration of generalized hypergeometric functions through precise remainder asymptotics

Acceleration of generalized hypergeometric functions through precise remainder asymptotics

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Article ID: iaor2012938
Volume: 59
Issue: 3
Start Page Number: 447
End Page Number: 485
Publication Date: Mar 2012
Journal: Numerical Algorithms
Authors:
Keywords: global convergence
Abstract:

We express the asymptotics of the remainders of the partial sums {s n } of the generalized hypergeometric function syntax error at token } equ1 through an inverse power series z n n λ c k n k equ2 , where the exponent λ 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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