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A Generalization of the Bernoulli Numbers

The Bernoulli numbers are among the most interesting and important number sequences in mathematics. It plays an important and quite mysterious role in various places like number theory, analysis and etc. In general, many existing generalizations of Bernoulli numbers { for example []20re based on con...

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Main Author: Bahari, Hasmiah
Format: Thesis
Language:English
English
Published: 2006
Online Access:http://psasir.upm.edu.my/id/eprint/5400/1/IPM_2006_4.pdf
http://psasir.upm.edu.my/id/eprint/5400/
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spelling my.upm.eprints.54002013-05-27T07:22:33Z http://psasir.upm.edu.my/id/eprint/5400/ A Generalization of the Bernoulli Numbers Bahari, Hasmiah The Bernoulli numbers are among the most interesting and important number sequences in mathematics. It plays an important and quite mysterious role in various places like number theory, analysis and etc. In general, many existing generalizations of Bernoulli numbers { for example []20re based on consideration of more general forms for the left side of the following equality }nB,21 ()Σ∞==−0!1expnnnntBtt or for some related functions. In this study, a generalization of Bernoulli numbers is offered by the use of their relations with Pascal’s triangle. The thesis begins with the generalization of Bernoulli numbers {} then a representation of is presented, followed by the proof of the main result for odd n case (even case of n was considered in ∞=1nnBnB[]2 ). Then special cases of Bernoulli numbers, namely when the initial sequence is an geometric or arithmetic sequence, are considered. In these special cases more detailed representations of are obtained. Then irreducibility problem over Z of polynomials closely related to is considered followed by solution of this problem for some values of . At the end some unsolved problems, with which we have come across in doing this thesis, over the field nBnBnZ are formulated. 2006 Thesis NonPeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/5400/1/IPM_2006_4.pdf Bahari, Hasmiah (2006) A Generalization of the Bernoulli Numbers. Masters thesis, Universiti Putra Malaysia. English
institution Universiti Putra Malaysia
building UPM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Putra Malaysia
content_source UPM Institutional Repository
url_provider http://psasir.upm.edu.my/
language English
English
description The Bernoulli numbers are among the most interesting and important number sequences in mathematics. It plays an important and quite mysterious role in various places like number theory, analysis and etc. In general, many existing generalizations of Bernoulli numbers { for example []20re based on consideration of more general forms for the left side of the following equality }nB,21 ()Σ∞==−0!1expnnnntBtt or for some related functions. In this study, a generalization of Bernoulli numbers is offered by the use of their relations with Pascal’s triangle. The thesis begins with the generalization of Bernoulli numbers {} then a representation of is presented, followed by the proof of the main result for odd n case (even case of n was considered in ∞=1nnBnB[]2 ). Then special cases of Bernoulli numbers, namely when the initial sequence is an geometric or arithmetic sequence, are considered. In these special cases more detailed representations of are obtained. Then irreducibility problem over Z of polynomials closely related to is considered followed by solution of this problem for some values of . At the end some unsolved problems, with which we have come across in doing this thesis, over the field nBnBnZ are formulated.
format Thesis
author Bahari, Hasmiah
spellingShingle Bahari, Hasmiah
A Generalization of the Bernoulli Numbers
author_facet Bahari, Hasmiah
author_sort Bahari, Hasmiah
title A Generalization of the Bernoulli Numbers
title_short A Generalization of the Bernoulli Numbers
title_full A Generalization of the Bernoulli Numbers
title_fullStr A Generalization of the Bernoulli Numbers
title_full_unstemmed A Generalization of the Bernoulli Numbers
title_sort generalization of the bernoulli numbers
publishDate 2006
url http://psasir.upm.edu.my/id/eprint/5400/1/IPM_2006_4.pdf
http://psasir.upm.edu.my/id/eprint/5400/
_version_ 1643823180214697984
score 13.211869

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