Numerical Algorithms

, Volume 59, Issue 3, pp 447–485 | Cite as

Acceleration of generalized hypergeometric functions through precise remainder asymptotics

Original Paper

Abstract

We express the asymptotics of the remainders of the partial sums {sn} of the generalized hypergeometric function \({\ensuremath{{}_{q+1}F_q\!\left(\left.\begin{smallmatrix}\alpha_1,\ldots,\alpha_{q+1}\\ \beta_1,\ldots,\beta_q\end{smallmatrix}\right|z\right)}}\) through an inverse power series \(z^n n^{\lambda} \sum \frac{c_k}{n^k}\), where the exponent λ and the asymptotic coefficients {ck} 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.

Keywords

Generalized hypergeometric functions Series acceleration Recurrence asymptotics 

Mathematics Subject Classifications (2010)

33C20 65B10 33F05 65D20 

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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of PhysicsAbilene Christian UniversityAbileneUSA
  2. 2.Max-Planck-Institut für GravitationsphysikHannoverGermany

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