Title: On the bounds for the spectral norms of geometric circulant matrices with generalized Jacobsthal and Jacobsthal Lucas numbers
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-21-00045; Volume 4 / Issue 1 / Year 2022, Pages 107-119
Document Type: Research Paper
aDepartment of Mathematics, Science and Art Faculty, Gaziantep University, Campus, 27310, Gaziantep, Turkey
Received: 23 June 2021, Accepted: 5 October 2021, Published: 24 December 2021.
Corresponding Author: Şükran Uygun (Email address: firstname.lastname@example.org)
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The study is about the different norms of geometric circulant matrices with the sequences called (s, t)-Jacobsthal, (s, t)-Jacobsthal Lucas and hyperharmonic Jacobsthal numbers. In the paper we obtain the upper and lower bounds for the spectral norms of geometric circulant matrix with the (s, t)-Jacobsthal and (s, t)-Jacobsthal Lucas numbers and also with hyperharmonic Jacobsthal numbers.
Keywords: Jacobsthal numbers, Jacobsthal Lucas numbers, hyperharmonic numbers, geometric circulant matrix, normReferences:
- M. Bahsi, On the norms of circulant matrices with the generalized Fibonacci and Lucas numbers, TWMS J. Pure Appl. Math. 6 (1), 84–92, 2015.
- M. Bahsi, On the norms of r-circulant matrices with the hyperharmonic numbers, J. Math Inequal. 10 (2), 445–458, 2016.
- M. Bahsi and S. Solak, On the norms of r-circulant matrices with the hyper-Fibonacci and Lucas numbers, J. Math Inequal. 8 (4), 693–705, 2014.
- A. Dil and I. Mezo, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comput. 217, 6011–6012, 2011.
- C. He, J. Ma, K. Zhang and Z. Wang, The upper bound estimation on the spectral norm r-circulant matrices with the Fibonacci and Lucas numbers, J. Inequal Appl. 72, 1–10, 2015.
- A. F. Horadam, Jacobsthal representation numbers, Fibonacci Q. 34 (1), 40–54, 1996.
- R. A. Horn and C. R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.
- C. Kizilates and N. Tuglu, On the bounds for the spectral norms of geometric circulant matrices, J. Inequal Appl. 312, 1–15, 2016.
- E. G. Kocer, T. Mansour and N. Tuglu, Norms of circulant and semicurculant matrices with Horadam’s numbers, Ars Combinatoria 85, 353–359, 2007.
- T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc., New York, 2001.
- R. Mathias, The spectral norm of nonnegative matrix, Linear Algebra Appl. 131, 269–284, 1990.
- R. Reams, Hadamard inverses square roots and products of almost semidefinite matrices, Linear Algebra Appl. 288, 35–43, 1999.
- S. Shen and J. Cen, On the bounds for the norms of r-circulant matrices with the Fibonocci and Lucas numbers, Appl. Math. Comput. 216, 2891–2897, 2010.
- S. Solak, On the norms of circulant matrices with the Fibonocci and Lucas numbers, Appl. Math. Comput. 160, 125–132, 2005.
- N. Tuglu and C. Kizilates, On the norms of some special matrices with the harmonic Fibonacci numbers, Gazi Univ. J. Sci. 28 (3), 447–501, 2015.
- N. Tuglu and C. Kizilates, On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers, J. Inequal Appl. 253, 1-11, 2015.
- N. Tuglu, C. Kizilateş and S. Kesim, On the harmonic and hyperharmonic Fibonacci numbers with the hyper-Fibonacci numbers, Adv. Differ. Equ. 297, 1–12, 2015.
- K. Uslu, N. Taşkara and Ş. Uygun, The relations among k-Fibonacci, k-Lucas and, generalized k-Fibonacci numbers and the spectral norms of the matrices of involving these numbers, Ars Combinatoria 102, 183–192, 2011.
- Ş. Uygun, The (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas sequences, Appl. Math. Sci. 70 (9), 3467–3476, 2015.
- Ş. Uygun, Some bounds for the norms of circulant matrices with the k-Jacobsthal and k-Jacobsthal Lucas numbers, Journal of Mathematics Research 8 (6), 133–138, 2016.
- Y. Yazlık and N. Taşkara, On the norms of an r-circulant matriceswith the generalized k-Horadam numbers, J. Inequal Appl. 394, 1–8, 2013.
- G. Zielke, Some remarks on matrix norms, condition numbers and error estimates for linear equations, Linear Algebra Appl. 110, 29–41, 1988.