Acceleration of generalized hypergeometric functions through precise remainder asymptotics

Abstract

We express the asymptotics of the remainders of the partial sums {s n } of the generalized hypergeometric function q+1 F q α 1 ,...,α q+1 β 1 ,...,β q z through an inverse power series z n n λ c k n k , 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.

On exact pricing of FX options in multivariate time-changed Lévy models

“…Usually, integral representations, polynomial approaches or the numerical solution of differential equations are used for the calculation of the generalized hypergeometric functions. For some new methods, including the remainder deriving, see the work by Willis (2012) and references there. Also, throughout this work we denote as B(x, y) and K γ (x) the beta function and the McDonald function, respectively.…”

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“…We could cover |z| ≫ 1 by evaluating p F q -series directly, but other methods are needed on the annulus surrounding the unit circle. Convergence accelerations schemes such as [69,7,56] can be effective, but the D-finite analytic continuation approach (with the singular expansion at z = 1) is likely better since effective error bounds are known and since complexity-reduction techniques apply. A study remains to be done.…”

Recursive determination of phase shifts for screened Coulomb potentials

“…The only transformation formula employed is the use of the linear transformation when | | z is approaching 1 from below. Recently, Willis [50] proposed an acceleration procedure through precise remainder asymptotics. He expressed the asymptotics of the remainders of the partial sums of the generalized hypergeometric function F p q through an inverse power series where the exponent λ and the asymptotic coefficients c k may be recursively computed to any desired order from the hypergeometric parameters and argument.…”

“…when |z| is approaching 1 from below. Recently, Willis [50] proposed an acceleration procedure through precise remainder asymptotics. He expressed the asymptotics of the remainders of the partial sums of the generalized hypergeometric function p F q through an inverse power series…”