On Generalized Fibonacci Numbers
Fibonacci numbers and their polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions, and by varying the recurrence relation and maintaining the initial conditions. In this paper, we introduce and derive various properties of $r...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Emrah Evren KARA
2020-12-01
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| Series: | Communications in Advanced Mathematical Sciences |
| Subjects: | |
| Online Access: | https://dergipark.org.tr/tr/download/article-file/1207040 |
| _version_ | 1797294235342340096 |
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| author | Isaac Owino Okoth Fidel Oduol |
| author_facet | Isaac Owino Okoth Fidel Oduol |
| author_sort | Isaac Owino Okoth |
| collection | DOAJ |
| description | Fibonacci numbers and their polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions, and by varying the recurrence relation and maintaining the initial conditions. In this paper, we introduce and derive various properties of $r$-sum Fibonacci numbers. The recurrence relation is maintained but initial conditions are varied. Among results obtained are Binet's formula, generating function, explicit sum formula, sum of first $n$ terms, sum of first $n$ terms with even indices, sum of first $n$ terms with odd indices, alternating sum of $n$ terms of $r-$sum Fibonacci sequence, Honsberger's identity, determinant identities and a generalized identity from which Cassini's identity, Catalan's identity and d'Ocagne's identity follow immediately. |
| first_indexed | 2024-03-07T21:27:18Z |
| format | Article |
| id | doaj.art-dd20c08f91984f36ac9591a6a9d27a3b |
| institution | Directory Open Access Journal |
| issn | 2651-4001 |
| language | English |
| last_indexed | 2024-03-07T21:27:18Z |
| publishDate | 2020-12-01 |
| publisher | Emrah Evren KARA |
| record_format | Article |
| series | Communications in Advanced Mathematical Sciences |
| spelling | doaj.art-dd20c08f91984f36ac9591a6a9d27a3b2024-02-27T04:36:36ZengEmrah Evren KARACommunications in Advanced Mathematical Sciences2651-40012020-12-013418620210.33434/cams.7710231225On Generalized Fibonacci NumbersIsaac Owino Okoth0Fidel Oduol1Maseno UniversityMaseno UniversityFibonacci numbers and their polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions, and by varying the recurrence relation and maintaining the initial conditions. In this paper, we introduce and derive various properties of $r$-sum Fibonacci numbers. The recurrence relation is maintained but initial conditions are varied. Among results obtained are Binet's formula, generating function, explicit sum formula, sum of first $n$ terms, sum of first $n$ terms with even indices, sum of first $n$ terms with odd indices, alternating sum of $n$ terms of $r-$sum Fibonacci sequence, Honsberger's identity, determinant identities and a generalized identity from which Cassini's identity, Catalan's identity and d'Ocagne's identity follow immediately.https://dergipark.org.tr/tr/download/article-file/1207040binet's formulafibonacci sequencegenerating functionr-shifted fibonacci sequence |
| spellingShingle | Isaac Owino Okoth Fidel Oduol On Generalized Fibonacci Numbers Communications in Advanced Mathematical Sciences binet's formula fibonacci sequence generating function r-shifted fibonacci sequence |
| title | On Generalized Fibonacci Numbers |
| title_full | On Generalized Fibonacci Numbers |
| title_fullStr | On Generalized Fibonacci Numbers |
| title_full_unstemmed | On Generalized Fibonacci Numbers |
| title_short | On Generalized Fibonacci Numbers |
| title_sort | on generalized fibonacci numbers |
| topic | binet's formula fibonacci sequence generating function r-shifted fibonacci sequence |
| url | https://dergipark.org.tr/tr/download/article-file/1207040 |
| work_keys_str_mv | AT isaacowinookoth ongeneralizedfibonaccinumbers AT fideloduol ongeneralizedfibonaccinumbers |