PreprintCC BY 4.0arXiv
The Levin Method for the Summation of One-dimensional and Multidimensional Infinite Series
Abstract
The Levin method transforms the evaluation of a highly oscillatory integral into the solution of a first-order linear ODE for a slowly varying auxiliary function. This ODE is typically approximated by collocation, after which the integral value is recovered from the auxiliary function at the endpoints. The present work develops a new extension of the Levin method for the summation of one-dimensional and multidimensional infinite oscillatory series. The summation problem is transformed into the solution of a functional equation involving transformed arguments of the unknown function. The resulting approach is particularly attractive in the multidimensional setting, where the range of existing numerical methods is relatively limited.
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BibTeX
@misc{levin2026levin,
title = {{The Levin Method for the Summation of One-dimensional and Multidimensional Infinite Series}},
author = {David Levin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06152},
eprint = {2609.06152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06152},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Levin Method for the Summation of One-dimensional and Multidimensional Infinite Series
AU - David Levin
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.06152
UR - https://arxiv.org/abs/2609.06152
AB - The Levin method transforms the evaluation of a highly oscillatory integral into the solution of a first-order linear ODE for a slowly varying auxiliary function. This ODE is typically approximated by collocation, after which the integral value is recovered from the auxiliary function at the endpoints. The present work develops a new extension of the Levin method for the summation of one-dimensional and multidimensional infinite oscillatory series. The summation problem is transformed into the solution of a functional equation involving transformed arguments of the unknown function. The resulting approach is particularly attractive in the multidimensional setting, where the range of existing numerical methods is relatively limited.
ER -