An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers
Let R=(R,+,·) be a commutative ring of characteristic m>0 (m may be equal to +∞) with unity e and zero 0. Given a positive integer n<m and the so-called n-symmetric set A=a1,a2,…,a2l-1,a2l such that al+i=ne-ai for each i=1,…,l, define the rth power sum Sr(A) as Sr(A)=∑i=12lair, for r=0,1,2,…....
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2017/9092515 |
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| _version_ | 1832564518356516864 |
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| author | Miomir Andjić Romeo Meštrović |
| author_facet | Miomir Andjić Romeo Meštrović |
| author_sort | Miomir Andjić |
| collection | DOAJ |
| description | Let R=(R,+,·) be a commutative ring of characteristic m>0 (m may be equal to +∞) with unity e and zero 0. Given a positive integer n<m and the so-called n-symmetric set A=a1,a2,…,a2l-1,a2l such that al+i=ne-ai for each i=1,…,l, define the rth power sum Sr(A) as Sr(A)=∑i=12lair, for r=0,1,2,…. We prove that for each positive integer k there holds ∑i=02k-1(-1)i2k-1i22k-1-iniS2k-1-i(A)=0. As an application, we obtain two new Pascal-like identities for the sums of powers of the first n-1 positive integers. |
| format | Article |
| id | doaj-art-077d318908454a109b370c879808a1b3 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-077d318908454a109b370c879808a1b32025-02-03T01:10:44ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2017-01-01201710.1155/2017/90925159092515An Identity in Commutative Rings with Unity with Applications to Various Sums of PowersMiomir Andjić0Romeo Meštrović1Faculty for Information Technology, University “Mediterranean”, Vaka Djurovića BB, Podgorica, MontenegroMaritime Faculty Kotor, University of Montenegro, Dobrota 36, 85330 Kotor, MontenegroLet R=(R,+,·) be a commutative ring of characteristic m>0 (m may be equal to +∞) with unity e and zero 0. Given a positive integer n<m and the so-called n-symmetric set A=a1,a2,…,a2l-1,a2l such that al+i=ne-ai for each i=1,…,l, define the rth power sum Sr(A) as Sr(A)=∑i=12lair, for r=0,1,2,…. We prove that for each positive integer k there holds ∑i=02k-1(-1)i2k-1i22k-1-iniS2k-1-i(A)=0. As an application, we obtain two new Pascal-like identities for the sums of powers of the first n-1 positive integers.http://dx.doi.org/10.1155/2017/9092515 |
| spellingShingle | Miomir Andjić Romeo Meštrović An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers Discrete Dynamics in Nature and Society |
| title | An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers |
| title_full | An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers |
| title_fullStr | An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers |
| title_full_unstemmed | An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers |
| title_short | An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers |
| title_sort | identity in commutative rings with unity with applications to various sums of powers |
| url | http://dx.doi.org/10.1155/2017/9092515 |
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