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Acceleration of sequences with transformations involving hypergeometric functions

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Abstract

We introduce a novel class of non-linear transformations of sequences that can accelerate both linear and logarithmic convergence and perform the summation of divergent series. Those transformations naturally arise when the ratio of successive forward differences of the original sequence are approximated by rational functions. The transformed sequences are thus expressible in terms of generalized hypergeometric functions, whose parameters depend on the original sequence. We explicitly derive analytical expressions for the lowest order transformations, present an alternative derivation using continued fractions, investigate convergence and accelerative properties, and derive the kernels of the transformations. The lowest order transformation is equivalent to Aitken’s Δ2 process, while, in the following order, the transformation specialized to logarithmic convergence is equivalent to Lubkin-W transformation. Numerical examples that illustrate the power and limitations of iterations or combinations of hypergeometric transformations are also investigated.

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Acknowledgements

The author would like to thank Prof. Claude Brezinski and the anonymous reviewers for invaluable suggestions.

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  1. SRT-ES Ministério do Trabalho e Previdência, Rua Pietrangelo de Biase, 56 - Centro, Vitória, ES, 29010-190, Brazil

    Rafael Tristão Pepino

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  1. Rafael Tristão Pepino
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Correspondence to Rafael Tristão Pepino.

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Pepino, R.T. Acceleration of sequences with transformations involving hypergeometric functions. Numer Algor 92, 893–915 (2023). https://doi.org/10.1007/s11075-022-01334-7

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