Identities relating six members of the Fibonacci family of sequences
In this paper, we prove several identities each relating a sum of products of three terms coming from different members of the Fibonacci family of sequences with a comparable sum whose terms come from three other sequences. These identities are obtained as special cases of formulas relating two line...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2022-02-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/4915 |
| _version_ | 1797205803833229312 |
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| author | R. Frontczak T. Goy M. Shattuck |
| author_facet | R. Frontczak T. Goy M. Shattuck |
| author_sort | R. Frontczak |
| collection | DOAJ |
| description | In this paper, we prove several identities each relating a sum of products of three terms coming from different members of the Fibonacci family of sequences with a comparable sum whose terms come from three other sequences. These identities are obtained as special cases of formulas relating two linear combinations of products of three generalized Fibonacci or Lucas sequences. The latter formulas are in turn obtained from a more general generating function result for the product of three terms coming from second-order linearly recurrent sequences with arbitrary initial values. We employ algebraic arguments to establish our results, making use of the Binet-like formulas of the underlying sequences. Among the sequences for which the aforementioned identities are found include the Fibonacci, Pell, Jacobsthal and Mersenne numbers, along with their associated Lucas companion sequences. |
| first_indexed | 2024-04-24T08:56:56Z |
| format | Article |
| id | doaj.art-bdec8b4da357479397e2af8d61519eae |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:56:56Z |
| publishDate | 2022-02-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-bdec8b4da357479397e2af8d61519eae2024-04-16T07:10:59ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102022-02-0114161910.15330/cmp.14.1.6-194273Identities relating six members of the Fibonacci family of sequencesR. Frontczak0T. Goy1https://orcid.org/0000-0002-6212-3095M. Shattuck2Landesbank Baden-Württemberg, 70173, Stuttgart, GermanyVasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, UkraineUniversity of Tennessee, 37996 Knoxville, TN, USAIn this paper, we prove several identities each relating a sum of products of three terms coming from different members of the Fibonacci family of sequences with a comparable sum whose terms come from three other sequences. These identities are obtained as special cases of formulas relating two linear combinations of products of three generalized Fibonacci or Lucas sequences. The latter formulas are in turn obtained from a more general generating function result for the product of three terms coming from second-order linearly recurrent sequences with arbitrary initial values. We employ algebraic arguments to establish our results, making use of the Binet-like formulas of the underlying sequences. Among the sequences for which the aforementioned identities are found include the Fibonacci, Pell, Jacobsthal and Mersenne numbers, along with their associated Lucas companion sequences.https://journals.pnu.edu.ua/index.php/cmp/article/view/4915horadam sequencefibonacci sequencelucas sequencepell sequencejacobsthal sequencegibonacci sequencegenerating function |
| spellingShingle | R. Frontczak T. Goy M. Shattuck Identities relating six members of the Fibonacci family of sequences Karpatsʹkì Matematičnì Publìkacìï horadam sequence fibonacci sequence lucas sequence pell sequence jacobsthal sequence gibonacci sequence generating function |
| title | Identities relating six members of the Fibonacci family of sequences |
| title_full | Identities relating six members of the Fibonacci family of sequences |
| title_fullStr | Identities relating six members of the Fibonacci family of sequences |
| title_full_unstemmed | Identities relating six members of the Fibonacci family of sequences |
| title_short | Identities relating six members of the Fibonacci family of sequences |
| title_sort | identities relating six members of the fibonacci family of sequences |
| topic | horadam sequence fibonacci sequence lucas sequence pell sequence jacobsthal sequence gibonacci sequence generating function |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/4915 |
| work_keys_str_mv | AT rfrontczak identitiesrelatingsixmembersofthefibonaccifamilyofsequences AT tgoy identitiesrelatingsixmembersofthefibonaccifamilyofsequences AT mshattuck identitiesrelatingsixmembersofthefibonaccifamilyofsequences |