Acceleration of generalized hypergeometric functions through precise remainder asymptotics

TL;DR

This paper develops a new asymptotic expansion for the remainders of generalized hypergeometric functions, enabling an effective series acceleration method for precise computation even at branch points.

Contribution

It introduces a recursive method to compute asymptotic coefficients and exponents, leading to a novel series acceleration technique applicable to complex parameters and branch points.

Findings

Effective for moderate parameters with fixed-precision C implementation

Accurately determines convergence and estimates errors for larger parameters

Enables precise computation of hypergeometric functions at challenging points

Abstract

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.