The Hadamard Products for Bi-periodic Fibonacci and Bi-periodic Lucas Generating Matrices

Yıl 2018, Cilt: 6 Sayı: 2, 272 - 278, 15.10.2018

Arzu Coşkun , Necati Taşkara

Öz

In this paper, firstly, we define the $Q_{q}$ \textit{-generating matrix} for bi-periodic Fibonacci polynomial. And we give $n$th power, determinant and some properties of the bi-periodic Fibonacci polynomial by considering this matrix representation. Also, we introduce the Hadamard products for bi-periodic Fibonacci $Q_{q}^{n}$ generating matrix and bi-periodic Lucas $Q_{l}^{n}$ generating matrix of which entries are bi-periodic Fibonacci and Lucas numbers. Then, we investigate some properties of these products.

Anahtar Kelimeler

bi-periodic Fibonacci numbers, bi-periodic Lucas numbers, Fibonacci Q matrix, matrix method, bi-periodic Fibonacci polynomial, generating matrix

Kaynakça

• [1] Bilgici G., Two generalizations of Lucas sequence, Applied Mathematics and Computation, 245 (2014), 526-538.
• [2] Cerda-Morales G., Matrix representation of the q-Jacobsthal numbers, Proyecciones (Antofagasta), 31.4 (2012), 345-354.
• [3] Coskun A., Taskara N., A note on the bi-periodic Fibonacci and Lucas matrix sequences, Applied Mathematics and Computation, 320 (2018), 400-406.
• [4] Coskun A., Taskara N., Generating matrix of the bi-periodic Lucas numbers, AIP Conference Proceedings Vol:1863, No.1 (2017), 310004.
• [5] Edson M., Yayenie O., A new Generalization of Fibonacci sequence and Extended Binet’s Formula, Integers, 9.6 (2009),639-654.
• [6] Falcon S., Plaza A., On the Fibonacci k-numbers, Chaos, Solitons & Fractals, 32.5 (2007), 1615-1624.
• [7] Gulec H.H., Taskara N., On the (s; t)-Pell and (s; t) -Pell-Lucas sequences and their matrix representations, Applied Mathematics Letters, 25.10 (2012), 1554-1559.
• [8] Hoggatt V.E., Fibonacci and Lucas numbers, Houghton Mifflin, 1969.
• [9] Koshy T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, NY, 2001.
• [10] Nalli A., On the Hadamard Product of Fibonacci Qn matrix and Fibonacci Q􀀀n matrix, Int. J. Contemp. Math. Sciences, 1.16 (2006), 753-761.
• [11] Ocal A.A., Tuglu N., Altinisik E., On the representation of k-generalized Fibonacci and Lucas numbers, Applied Mathematics and Computation, 170.1 (2005), 584-596.
• [12] Sylvester J.R., Fibonacci Properties by Matrix Method, Mathematical Gazette, 63 (1979), 188-191.
• [13] Tasci D., On the Hadamard Products of its Adjoint matrix with a square matrix, Selcuk University Journal of Science, 1.17 (2000), 43-53.
• [14] Uslu K., Uygun S., The (s; t) Jacobsthal and (s; t) Jacobsthal-Lucas Matrix Sequences, ARS Combinatoria, 108 (2013), 13-22.
• [15] Vajda S., Fibonacci & Lucas numbers and the golden section. Theory and Applications, Ellis Horwood Series, Mathematics and Applications, 1989.
• [16] Yazlik Y., Taskara N., Uslu K., Yilmaz N., The Generalized (s; t)-Sequence and its Matrix Sequence, AIP Conf. Proc. Vol:1389, No.1 (2011), 381-384.
• [17] Yilmaz N., Coskun A., Taskara N., On properties of bi-periodic Fibonacci and Lucas polynomials, AIP Conference Proceedings Vol:1863, No.1 (2017), 310002.