Abstract
In this paper, we define the bi-periodic Fibonacci matrix sequence that represent bi-periodic Fibonacci numbers. Then, we investigate generating function, Binet formula and summations of bi-periodic Fibonacci matrix sequence. After, we say that some behaviours of bi-periodic Fibonacci numbers can be obtained via the properties of this new matrix sequence. Finally, we express that well-known matrix sequences such as Fibonacci, Pell, k-Fibonacci matrix sequences are special cases of this generalized matrix sequence.
Keywords
Bi-periodic Fibonacci matrix sequence bi-periodic Fibonacci numbers Binet formula generating function
References
- Bilgici G., Two generalizations of Lucas sequence, Applied Mathematics and Computation, 245 (2014), 526-538.
- Civciv H., Türkmen R., On the (s,t)-Fibonacci and Fibonacci matrix sequences, Ars Combinatoria, 87 (2008), 161-173.
- Edson M., Yayanie O., A new Generalization of Fibonacci sequence and Extended Binet's Formula, Integers, 9 (2009), 639-654.
- Falcon S., Plaza A., On the Fibonacci k-numbers, Chaos, Solitons & Fractal, 32 (2007), 1615-1624.
- Gulec H.H., Taskara N., On the (s,t)-Pell and (s,t)-Pell-Lucas sequences and their matrix representations, Applied Mathematics Letter, 25 (2012), 1554-1559.
- Horadam A.F., A generalized Fibonacci sequence, Math. Mag., 68 (1961), 455-459.
- Koshy T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, NY, 2001.
- Marek-Crnjac L, The mass spectrum of high energy elementary particles via El Naschie's golden mean nested oscillators, the Dunkerly-Southwell eigenvalue theorems and KAM, Chaos, Solutions & Fractals, 18(1) (2003), 125-133.
- Ocal A.A., Tuglu N., Altinisik E., On the representation of k-generalized Fibonacci and Lucas numbers, Applied Mathematics and Computations, 170 (1) (2005), 584-596.
- Panario D., Sahin M., Wang Q., Non-homogeneous conditional recurrences, Linear and Multilinear Algebra, 66 (10) (2018), 2089-2099.
- Sahin M., The generating function of a family of the sequences in terms of the continuant, Applied Mathematics and Computations, 217 (12) (2011), 5416-5420.
- Tan E., Ekin A.B., Some identities on conditional sequences by using matrix method, Miskolc Mathematical Notes, 18 (1) (2017), 469-477.
- Tan E., On bi-periodic Fibonacci and Lucas numbers by matrix method, Ars Combinatoria, 133 (2017), 107-113.
- Tasci D., Firengiz M.C., Incomplete Fibonacci and Lucas p-numbers, Mathematical and Computer Modelling, 52(9) (2010), 1763-1770.
- Uslu K., Uygun S., The (s,t) Jacobsthal and (s,t) Jacobsthal-Lucas Matrix Sequences, Ars Combinatoria, 108 (2013), 13-22.
- Vajda S., Fibonacci & Lucas numbers and the golden section. Theory and Applications, Ellis Horwood Limited, 1989.
- Yazlik Y., Taskara N., A note on generalized k-Horadam sequence, Computers and Mathematics with Applications, 63 (2012), 36-41.
- Yazlik Y., Taskara N., Uslu K., Yilmaz N., The Generalized (s,t)-Sequence and its Matrix Sequence, American Institute of Physics (AIP) Conf. Proc., 1389 (2012), 381-384.
Abstract
References
- Bilgici G., Two generalizations of Lucas sequence, Applied Mathematics and Computation, 245 (2014), 526-538.
- Civciv H., Türkmen R., On the (s,t)-Fibonacci and Fibonacci matrix sequences, Ars Combinatoria, 87 (2008), 161-173.
- Edson M., Yayanie O., A new Generalization of Fibonacci sequence and Extended Binet's Formula, Integers, 9 (2009), 639-654.
- Falcon S., Plaza A., On the Fibonacci k-numbers, Chaos, Solitons & Fractal, 32 (2007), 1615-1624.
- Gulec H.H., Taskara N., On the (s,t)-Pell and (s,t)-Pell-Lucas sequences and their matrix representations, Applied Mathematics Letter, 25 (2012), 1554-1559.
- Horadam A.F., A generalized Fibonacci sequence, Math. Mag., 68 (1961), 455-459.
- Koshy T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, NY, 2001.
- Marek-Crnjac L, The mass spectrum of high energy elementary particles via El Naschie's golden mean nested oscillators, the Dunkerly-Southwell eigenvalue theorems and KAM, Chaos, Solutions & Fractals, 18(1) (2003), 125-133.
- Ocal A.A., Tuglu N., Altinisik E., On the representation of k-generalized Fibonacci and Lucas numbers, Applied Mathematics and Computations, 170 (1) (2005), 584-596.
- Panario D., Sahin M., Wang Q., Non-homogeneous conditional recurrences, Linear and Multilinear Algebra, 66 (10) (2018), 2089-2099.
- Sahin M., The generating function of a family of the sequences in terms of the continuant, Applied Mathematics and Computations, 217 (12) (2011), 5416-5420.
- Tan E., Ekin A.B., Some identities on conditional sequences by using matrix method, Miskolc Mathematical Notes, 18 (1) (2017), 469-477.
- Tan E., On bi-periodic Fibonacci and Lucas numbers by matrix method, Ars Combinatoria, 133 (2017), 107-113.
- Tasci D., Firengiz M.C., Incomplete Fibonacci and Lucas p-numbers, Mathematical and Computer Modelling, 52(9) (2010), 1763-1770.
- Uslu K., Uygun S., The (s,t) Jacobsthal and (s,t) Jacobsthal-Lucas Matrix Sequences, Ars Combinatoria, 108 (2013), 13-22.
- Vajda S., Fibonacci & Lucas numbers and the golden section. Theory and Applications, Ellis Horwood Limited, 1989.
- Yazlik Y., Taskara N., A note on generalized k-Horadam sequence, Computers and Mathematics with Applications, 63 (2012), 36-41.
- Yazlik Y., Taskara N., Uslu K., Yilmaz N., The Generalized (s,t)-Sequence and its Matrix Sequence, American Institute of Physics (AIP) Conf. Proc., 1389 (2012), 381-384.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Submission Date | May 1, 2018 |
| Acceptance Date | April 6, 2019 |
| Publication Date | August 1, 2019 |
| DOI | https://doi.org/10.31801/cfsuasmas.571975 |
| IZ | https://izlik.org/JA28SS75DG |
| Published in Issue | Year 2019 Volume: 68 Issue: 2 |
Cite
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Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
