On Generalized Jacobsthal and Jacobsthal–Lucas Numbers
Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobstha...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Sciendo
2022-09-01
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| Series: | Annales Mathematicae Silesianae |
| Subjects: | |
| Online Access: | https://doi.org/10.2478/amsil-2022-0011 |
| _version_ | 1811247112437891072 |
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| author | Bród Dorota Michalski Adrian |
| author_facet | Bród Dorota Michalski Adrian |
| author_sort | Bród Dorota |
| collection | DOAJ |
| description | Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal–Lucas sequence. We give generating functions, Binet’s formulas for these numbers. Moreover, we obtain some identities, among others Catalan’s, Cassini’s identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal–Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal–Lucas numbers. Additionally, we give a matrix representation of the presented numbers. |
| first_indexed | 2024-04-12T15:03:26Z |
| format | Article |
| id | doaj.art-92ea4711016f47138b945d15a9f0b935 |
| institution | Directory Open Access Journal |
| issn | 2391-4238 |
| language | English |
| last_indexed | 2024-04-12T15:03:26Z |
| publishDate | 2022-09-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Annales Mathematicae Silesianae |
| spelling | doaj.art-92ea4711016f47138b945d15a9f0b9352022-12-22T03:28:01ZengSciendoAnnales Mathematicae Silesianae2391-42382022-09-0136211512810.2478/amsil-2022-0011On Generalized Jacobsthal and Jacobsthal–Lucas NumbersBród Dorota0Michalski Adrian1Rzeszow University of Technology, Faculty of Mathematics and Applied Physics, Department of Discrete Mathematics, al. Powstańców Warszawy 8, 35-959Rzeszów, PolandRzeszow University of Technology, Faculty of Mathematics and Applied Physics, Department of Discrete Mathematics, al. Powstańców Warszawy 8, 35-959Rzeszów, PolandJacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal–Lucas sequence. We give generating functions, Binet’s formulas for these numbers. Moreover, we obtain some identities, among others Catalan’s, Cassini’s identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal–Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal–Lucas numbers. Additionally, we give a matrix representation of the presented numbers.https://doi.org/10.2478/amsil-2022-0011jacobsthal numbersjacobsthal–lucas numbersgeneralized jacobsthal numbersbinet formula11b3711c2015b36 |
| spellingShingle | Bród Dorota Michalski Adrian On Generalized Jacobsthal and Jacobsthal–Lucas Numbers Annales Mathematicae Silesianae jacobsthal numbers jacobsthal–lucas numbers generalized jacobsthal numbers binet formula 11b37 11c20 15b36 |
| title | On Generalized Jacobsthal and Jacobsthal–Lucas Numbers |
| title_full | On Generalized Jacobsthal and Jacobsthal–Lucas Numbers |
| title_fullStr | On Generalized Jacobsthal and Jacobsthal–Lucas Numbers |
| title_full_unstemmed | On Generalized Jacobsthal and Jacobsthal–Lucas Numbers |
| title_short | On Generalized Jacobsthal and Jacobsthal–Lucas Numbers |
| title_sort | on generalized jacobsthal and jacobsthal lucas numbers |
| topic | jacobsthal numbers jacobsthal–lucas numbers generalized jacobsthal numbers binet formula 11b37 11c20 15b36 |
| url | https://doi.org/10.2478/amsil-2022-0011 |
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