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 F-2(1)(a,b; ...
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 F-2(1)(a,b; c: z) is well suited to illuminate problems of that kind. Sequence transformations perform quite well fur most parameters and arguments. If. however, the third parameter c of a nonterminating hypergeometric series F-2(1) 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 kj 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 F-2(1) (a. b: c: z) are derived. (C) 2001 Elsevier Science B.V. All rights reserved.
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