Fast and stable rational approximation of generalized hypergeometric functions

TL;DR

This paper introduces efficient, stable recurrence relations for rational approximations of generalized hypergeometric functions, improving computational stability and confirming the superiority of factorial Levin-type transformations over Drummond transformations.

Contribution

Derives new recurrence relations for rational approximations of ${}_pF_q$ functions that are more stable and computationally efficient than previous methods.

Findings

Recurrence relations require $ ext{O}[ ext{max}\{p,q ight ext{(n+k)}]$ flops.

Numerical evidence shows increased stability of the new recurrence relations.

Theoretical analysis confirms the superiority of factorial Levin-type transformation over Drummond transformation.

Abstract

Rational approximations of generalized hypergeometric functions ${}_pF_q$ of type $(n+k,k)$ are constructed by the Drummond and factorial Levin-type sequence transformations. We derive recurrence relations for these rational approximations that require $\mathcal{O}[\max\{p,q\}(n+k)]$ flops. These recurrence relations come in two forms: for the successive numerators and denominators; and, for an auxiliary rational sequence and the rational approximations themselves. Numerical evidence suggests that these recurrence relations are much more stable than the original formul\ae~for the Drummond and factorial Levin-type sequence transformations. Theoretical results on the placement of the poles of both transformations confirm the superiority of factorial Levin-type transformation over the Drummond transformation.