Irregular Input Data in Convergence Acceleration and Summation Processes: General Considerations and Some Special Gaussian Hypergeometric Series as Model Problems

TL;DR

The paper addresses how irregular leading terms in slowly convergent or divergent sequences can undermine convergence-acceleration techniques. It develops a formal framework for path choices in transformation tables and demonstrates, using model problems centered on the Gaussian hypergeometric function ${}_2F_1(a,b;c;z)$, that excluding irregular leading coefficients can restore reliable acceleration and summation. It provides guidance on when analytic continuation or recurrence relations are preferable and furnishes new recurrence formulas for ${}_2F_1$ to aid computation. The work has practical implications for evaluating special functions and summing divergent perturbation series, including the infinite coupling limit $k_3$ of the sextic anharmonic oscillator, and it highlights the need for multiple, cross-validated transformations in numerical practice.

Abstract

Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain asymptotic regularities normally do exist, but the leading elements of a sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1 (a, b; c; z) is well suited to illuminate problems of that kind. Sequence transformations perform quite well for most parameters and arguments. If, however, the third parameter $c$ of a nonterminating hypergeometric series 2F1 is a negative real number, the terms initially grow in magnitude like the terms of a mildly divergent series. The use of the leading terms of such a series as input data leads to unreliable and even completely nonsensical results. In contrast, sequence transformations produce good results if the leading irregular terms are excluded from the transformation process. Similar problems occur also in perturbation expansions. For example, summation results for the infinite coupling limit k_3 of the sextic anharmonic oscillator can be improved considerably by excluding the leading terms from the transformation process. Finally, numerous new recurrence formulas for the 2F1 (a, b; c; z) are derived.