Relation of the Cyclotomic Equation with the Harmonic and Derived Series

We associate some (old) convergent series related to definite integrals with the cyclotomic equation x (m) − 1 = 0, for several natural numbers m; for example, for m = 3, x (3) − 1 = (x − 1)(1 + x + x (2)) leads to [Formula: see text]. In some cases, we express the results in terms of the Dirichlet...

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Detalles Bibliográficos
Autores principales: Boya, Luis J., Rivera, Cristian
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Hindawi Publishing Corporation 2015
Materias:
Research Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4320904/
https://www.ncbi.nlm.nih.gov/pubmed/25688380
http://dx.doi.org/10.1155/2015/950521
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author Boya, Luis J.
Rivera, Cristian
author_facet Boya, Luis J.
Rivera, Cristian
author_sort Boya, Luis J.
collection PubMed
description We associate some (old) convergent series related to definite integrals with the cyclotomic equation x (m) − 1 = 0, for several natural numbers m; for example, for m = 3, x (3) − 1 = (x − 1)(1 + x + x (2)) leads to [Formula: see text]. In some cases, we express the results in terms of the Dirichlet characters. Generalizations for arbitrary m are well defined but do imply integrals and/or series summations rather involved.
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spelling pubmed-43209042015-02-16 Relation of the Cyclotomic Equation with the Harmonic and Derived Series Boya, Luis J. Rivera, Cristian ScientificWorldJournal Research Article We associate some (old) convergent series related to definite integrals with the cyclotomic equation x (m) − 1 = 0, for several natural numbers m; for example, for m = 3, x (3) − 1 = (x − 1)(1 + x + x (2)) leads to [Formula: see text]. In some cases, we express the results in terms of the Dirichlet characters. Generalizations for arbitrary m are well defined but do imply integrals and/or series summations rather involved. Hindawi Publishing Corporation 2015 2015-01-22 /pmc/articles/PMC4320904/ /pubmed/25688380 http://dx.doi.org/10.1155/2015/950521 Text en Copyright © 2015 L. J. Boya and C. Rivera. https://creativecommons.org/licenses/by/3.0/ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
spellingShingle Research Article
Boya, Luis J.
Rivera, Cristian
Relation of the Cyclotomic Equation with the Harmonic and Derived Series
title Relation of the Cyclotomic Equation with the Harmonic and Derived Series
title_full Relation of the Cyclotomic Equation with the Harmonic and Derived Series
title_fullStr Relation of the Cyclotomic Equation with the Harmonic and Derived Series
title_full_unstemmed Relation of the Cyclotomic Equation with the Harmonic and Derived Series
title_short Relation of the Cyclotomic Equation with the Harmonic and Derived Series
title_sort relation of the cyclotomic equation with the harmonic and derived series
topic Research Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4320904/
https://www.ncbi.nlm.nih.gov/pubmed/25688380
http://dx.doi.org/10.1155/2015/950521
work_keys_str_mv AT boyaluisj relationofthecyclotomicequationwiththeharmonicandderivedseries
AT riveracristian relationofthecyclotomicequationwiththeharmonicandderivedseries

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