ON THE NORM OF CIRCULANT MATRICES VIA GENERALIZED TETRANACCI NUMBERS

Year 2019, Volume: 6 Issue: 2, 562 - 572, 26.12.2019

Fatma Yeşil Baran Tevfik Yetiş

https://doi.org/10.35193/bseufbd.662239

Cited By: 1

Abstract

In this study, the sum of first n terms of this series is formulated by obtaining the Binet formula for the generalized Tetranacci sequence 〖(T〗_n )_(n∈N), whose initial values are T_0=a,〖 T〗_1=b,〖 T〗_2=c,T_3=d and defined by the
T_n=pT_(n-1)+qT_(n-2)+rT_(n-3)+sT_(n-4)
recurrence relation for n≥4. The generating function is obtained for generalized Tetranacci number sequence. In addition, some matrix norms are calculated for the circulant matrices consisting of elements of the generalized Tetranacci number sequence.

Keywords

Circulant Matrix

References

• [1] Feinberg, M. (1963). Fibonacci-tribonacci. The Fibonacci Quarterly. 1(1), 71-74.
• [2] Waddill, M. E. (1992). The Tetranacci sequence and generalizations. The Fibonacci Quarterly, 30(1), 9-20.
• [3] Lind, D. A. (1970). A Fibonacci circulant. The Fibonacci Quarterly, 8(5), 449-455.
• [4] Davis, P. J. 1979, Circulant Matrices, John Wiley and Sons, New York.
• [5] Öcal, A. A., Tuglu, N., & Altinişik, E. (2005). On the representation of k-generalized Fibonacci and Lucas numbers. Applied mathematics and computation, 170(1), 584-596.
• [6] Alptekin, E. G. (2005). Pell, Pell-Lucas ve Modified Pell sayıları ile tanımlı circulant ve semicirculant matrisler (Doctoral dissertation, Selçuk Üniversitesi Fen Bilimleri Enstitüsü).
• [7] Solak, S. (2005). On the norms of circulant matrices with the Fibonacci and Lucas numbers. Applied Mathematics and Computation, 160(1), 125-132.
• [8] Kocer, E. G., Mansour, T., & Tuglu, N. (2007). Norms of circulant and semicirculant matrices with Horadam's numbers. Ars Combinatoria, 85, 353-359.
• [9] Shen, S. Q., & Cen, J. M. (2010). On the spectral norms of r-circulant matrices with the k-Fibonacci and k-Lucas numbers. Int. J. Contemp. Math. Sciences, 5(12), 569-578.
• [10] Bahsi, M., & Solak, S. (2014). On the norms of r-circulant matrices with the hyper-Fibonacci and Lucas numbers. J. Math. Inequal, 8(4), 693-705.
• [11] Tuglu, N., & Kızılateş, C. (2015). On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers. Journal of Inequalities and Applications, 2015(1), 253.
• [12] Kızılateş, C., & Tuglu, N. (2018). On the Norms of Geometric and Symmetric Geometric Circulant Matrices with the Tribonacci Number. Gazi University Journal of Science, 31(2), 555-567.
• [13] Kızılateş, C., & Tuglu, N. (2016). On the bounds for the spectral norms of geometric circulant matrices. Journal of Inequalities and Applications, 2016(1), 312.
• [14] Tuglu, N., & Kızılateş, C. (2015). On the Norms of Some Special Matrices With the Harmonic Fibonacci Numbers. Gazi University Journal of Science, 28(3), 497-501.
• [15] Kızılateş, C. (2017). On the Quadra Lucas-Jacobsthal Numbers. Karaelmas Science and Engineering Journal, 7(2), 619-621.
• [16] Polatlı, E. On The Bounds For The Spectral Norms Of . r-circulant Matrices With a Type of Catalan Triangle Numbers. Journal of Science and Arts, 48(3), 2019.
• [17] Bahşi, M. (2015). On the Norms of Circulant Matrices with the Generalized Fibonacci and Lucas Numbers. TWMS J. Pure Appl. Math. 6(1), 84-92.
• [18] Özkoç, A. Ardıyok, E. (2016). Circulant and Negacyclic Matrices Via Tetranacci Numbers. Honam Mathematical J. 38(4), 725-738.
• [19] Tascı, D., & Acar, H. (2017). Gaussian tetranacci numbers. Communications in Mathematics ans Applications, 8(3), 379-386.
• [20] Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3(3), 161-176.
• [21] Karner, H., Schneid, J., & Ueberhuber, C. W. (2003). Spectral decomposition of real circulant matrices. Linear Algebra and Its Applications, 367, 301-311.
• [22] Pollock, D. S. G. (2002). Circulant matrices and time-series analysis. International Journal of Mathematical Education in Science and Technology, 33(2), 213-230.
• [23] Zaveri M. N. , Patel, J. K. (2015). Patel J. K. Binet’s Formula for the Tetranacci Sequence. International Journal of Science and Research (IJSR), 78-96.
• [24] Spickerman, W.R. (1982). Binet's formula for the Tribonacci sequence, The Fibonacci Quarterly. 20 (2), 118-120.