Abstract
In this work, we study the \(r\)-circulant matrix \( C_r = Circ_r(c_0, c_1,c_2,...,c_{n-1})\) such that the entries of \(C_r \) are \(c_i=M_{k,a+ib}\) or \(c_i=R_{k,a+ib}\), where \(M_{k,a+ib}\) and \(R_{k,a+ib}\) are \(k\)-Mersenne and \(k\)-Mersenne–Lucas numbers, respectively. We obtain the eigenvalues and determinants for the matrices and some important identities for the \(k\)-Mersenne and \(k\)-Mersenne–Lucas numbers. Furthermore, we find norms and bounds estimation for the spectral norm for these \(r\)-circulant matrices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. F. Horadam, “Basic properties of a certain generalized sequence of numbers,” Fibonacci Quart. 3 (3), 161–176 (1965).
T. Koshy, Fibonacci and Lucas Numbers with Applications (A Wiley-Interscience, New York, 2001).
V. E. Hoggatt, Fibonacci and Lucas numbers (MA: Houghton Mifflin, 1969).
K. Prasad and H. Mahato, “Cryptography using generalized Fibonacci matrices with Affine–Hill cipher,” J. Discrete Math. Sci. Cryptogr. 25 (8), 2341–2352 (2022).
S. Çelik, İ. Durukan, and E. Özkan, “New recurrences on Pell numbers, Pell–Lucas numbers, Jacobsthal numbers, and Jacobsthal–Lucas numbers,” Chaos Solitons Fractals 150, 111173 (2021).
E. Özkan and M. Taştan, “A new families of Gauss k-Jacobsthal numbers and Gauss k-Jacobsthal–Lucas numbers and their polynomials,” J. Sci. Arts 20 (4), 893–908 (2020).
A. Shannon and L. Bernstein, “The Jacobi–Perron algorithm and the algebra of recursive sequences,” Bull. Aust. Math. Soc. 8 (2), 261–277 973..
R. Dikici and E. Özkan, “An application of Fibonacci sequences in groups,” Appl. Math. Comput. 136 (2-3), 323–331 (2003).
P. Catarino, H. Campos, and P. Vasco, “On the Mersenne sequence,” Ann. Math. Inform. 46, 37–53 (2016).
N. Saba, A. Boussayoud, and K. Kanuri, “Mersenne Lucas numbers and complete homogeneous symmetric functions,” J. Math. Comput. Sci. 24 (2), 127–139 (2021).
K. Uslu and V. Deniz, “Some identities of k-Mersenne numbers,” Adv. Appl. Discrete Math. 18 (4), 413–423 (2017).
Y. Soykan, “A study on generalized Mersenne numbers,” Journal of Progressive Research in Mathematics 18, 90–108 (2021).
M. Chelgham and A. Boussayoud, “On the k-Mersenne–Lucas numbers,” Notes Number Theory Discrete Math. 1 (27), 7–13 (2021).
M. Kumari, J. Tanti, and K. Prasad, “On some new families of k-Mersenne and generalized k-Gaussian Mersenne numbers and their polynomials,” Contributions to Discrete Mathematics, 18 (2), (2023).
E. Özkan and M. Uysal, “Mersenne–Lucas hybrid numbers,” Math. Montisnigri 52, 17–29 (2021).
R. Cline, R. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices,” Linear Algebra Appl. 8 (1), 25–33 (1974).
M. Andrecut, “Applications of left circulant matrices in signal and image processing,” Modern Phys. Lett. B 22 (04), 231–241 (2008).
H.-J. Wittsack, A. M. Wohlschläger, E. K. Ritzl, R. Kleiser, M. Cohnen, R. J. Seitz, and U. Mödder, “CT-perfusion imaging of the human brain: advanced deconvolution analysis using circulant singular value decomposition,” Comput. Med. Imaging Graph. 32 (1), 67–77 (2008).
B. J. Olson, S. W. Shaw, C. Shi, C. Pierre, and R. G. Parker, “Circulant matrices and their application to vibration analysis,” Appl. Mech. Rev. 66 (4) (2014).
A. C. Wilde, “Differential equations involving circulant matrices,” Rocky Mountain J. Math. 13 (1), 1–13 (1983).
W. Chen, J. Lin, and C. Chen, “The method of fundamental solutions for solving exterior axisymmetric Helmholtz problems with high wave-number,” Adv. Appl. Math. Mech. 5 (4), 477–493 (2013).
S. Solak, “On the norms of circulant matrices with the Fibonacci and Lucas numbers,” Appl. Math. Comput. 160 (1), 125–132 (2005).
S. Shen and J. Cen, “On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers,” Appl. Math. Comput. 216 (10), 2891–2897 (2010).
S.-Q. Shen and J.-M. Cen, “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 (2010).
Y. Yu and Z. Jiang, “On the norms and spreads of Fermat, Mersenne and Gaussian Fibonacci RFMLR circulant matrices,” WSEAS Trans. Math. 15 (1), 34–43 (2016).
S. Uygun, “Some bounds for the norms of circulant matrices with the k-Jacobsthal and k-Jacobsthal Lucas numbers,” J. Math. Res. 8 (6), 133–138 (2016).
R. Türkmen and H. Gökbaş, “On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers,” J. Inequal. Appl. 2016 (1), 1–7 (2016).
B. Shi, “The spectral norms of geometric circulant matrices with the generalized k-Horadam numbers,” J. Inequal. Appl. 2018 (1), 1–9 (2018).
A. C. F. Bueno, “On r-circulant matrices with Horadam numbers having arithmetic indices,” Notes Number Theory Discrete Math. 26 (2), 177–197 (2020).
Acknowledgments
The authors are thankful to the anonymous referee for their valuable and constructive comments which helped substantially to improve the manuscript in its present form.
Funding
The first and second authors would like to thank the University Grant Commission (UGC), India, for the research fellowship (1057/CSIR-UGC NET JUNE 2018 and 1196/CSIR-UGC NET JUNE 2019).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kumari, M., Prasad, K., Özkan, E. et al. On the Norms and Eigenvalues of \(r\)-Circulant Matrices with \(k\)-Mersenne and \(k\)-Mersenne–Lucas Numbers. Math Notes 114, 522–535 (2023). https://doi.org/10.1134/S0001434623090225
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434623090225