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## On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers

#### Mustafa BAHŞİ [1] , Süleyman SOLAK [2]

In this paper, we compute the spectral norms of $r-$ circulant matrices with the hyper-Fibonacci and hyper-Lucas numbers of the forms $F_{r}=Circ-r(F_k^{(0)},F_k^{(1)},...,F_k^{(n-1)})$$L_r=Circ-r(L_{k}^{\left( 0\right) },L_{k}^{\left( 1\right) },...,L_{k}^{\left( n-1\right) })$ and their Hadamard and Kronecker products. For this, we firstly compute the spectral and Euclidean norms of circulant matrices of the forms $F=Circ(F_{k}^{\left( 0\right) }, F_{k}^{\left( 1\right) },... ,F_{k}^{\left( n-1\right) })$ and $L=Circ(L_{k}^{\left( 0\right) },L_{k}^{\left( 1\right) },...,L_{k}^{\left( n-1\right) })$. Moreover, we give some examples related to special cases of our results.

Circulant matrix, r-circulant matrix, hyper- Fibonacci numbers, hyper-Lucas numbers, Euclidean norm, spectral norm
• Bae, J., \textit{Circulant matrix factorization based on schur algorithm for designing optical multimirror filters,} Japanese Journal of Applied Physics \textbf{45(6A)}(2006), 5163--5168.
• Bah\c{s}i, M., \textit{On the norms of r-circulant matrices with the hyperharmonic numbers}, Journal of Mathematical Inequalities \textbf{10(2)}(2016), 445-458.
• Bah\c{s}i, M., \textit{On the norms of circulant matrices with the generalized Fibonacci and Lucas numbers,} TWMS J. Pure Appl. Math. \textbf{6(1)}(2015), 84-92.
• Bah\c{s}i, M., Mez\"{o}, I., Solak, S.,\textit{ A symmetric algorithm for hyper-Fibonacci and hyper-Lucas numbers}, Annales Mathematicae et Informaticae \textbf{43}(2014), 19--27.
• Bah\c{s}i, M., S. Solak, \textit{On the norms of r-circulant matrices with the hyper-Fibonacci and Lucas numbers}, Journal of Mathematical Inequalities \textbf{8(4)}(2014), 693-705.
• Bahsi, M., Solak, S., \textit{On the circulant matrices with arithmetic sequence}, Int. J. Cont. Math. Sciences \textbf{5(25)}(2010), 1213 -- 1222.
• Cao, N-N., Zhao, F-Z., \textit{Some Properties of Hyperfibonacci and Hyperlucas Numbers,}\ Journal of Integer Sequences \textbf{13}(2010), Article 10.0.8.
• Davis, P.J., \textit{Circulant Matrices,} Wiley, New York, Chichester, Brisbane, 1979.
• Dil, A., Mez\"{o}, I., \textit{A symmetric algorithm for hyperharmonic and Fibonacci numbers,}\ Appl. Math. Comp. \textbf{206}(2008), 942-951.
• Fischer, B., Modersitzki, J., \textit{Fast inversion of matrices arising in image processing,} Numer. Algorithms \textbf{22}(1999), 1--11.
• Georgiou, S.D., Kravvaritis, C., \textit{New Good Quasi-Cyclic Codes over GF(3),} Int. J. Algebra \textbf{1(1)}(2007), 11--24.
• Horn, R.A., Johnson, C.R., \textit{Matrix Analysis}, Cambridge University Press, Cambridge, 1985.
• Horn, R.A., Johnson, C.R., \textit{Topics in Matrix Analysis}, Cambridge University Press, Cambridge, 1991.
• Karner, H., Schneid, J., Ueberhuber, C.W.,\textit{Spectral Decomposition of Real Circulant Matrices,} Linear Algebra and Its Appl., \textbf{367}(2003), 301-311.
• K\i z\i late\c{s}, C., Tuglu, N., \textit{On the bounds for the spectral norms of geometric circulant matrices}, Journal of Inequalities and Applications (2016), 2016:312.
• Kocer, E.G., \textit{Circulant, negacyclic and semicirculant matrices with the modified Pell, Jacobsthal and Jacobsthal-Lucas numbers,} Hacettepe Journal of Mathematics and Statistics \textbf{36(2)}(2007), 133--142.
• Kocer, E.G., Mansour, T., Tu\u{g}lu, N., \textit{Norms of circulant and semicirculant matrices with Horadam's numbers,} Ars Combinatoria \textbf{85}(2007), 353-359.
• Tuglu, N., K\i z\i late\c{s}, C., \textit{On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers}, Journal of Inequalities and Applications (2015), 2015:253.
• Tuglu, N., K\i z\i late\c{s}, C., \textit{On the Norms of Some Special Matrices with the Harmonic Fibonacci Numbers,} Gazi University Journal of Science \textbf{28(3)}(2015), 497-501.
• \"{O}cal, A.A., Tu\u{g}lu, N., Alt\i n\i \c{s}\i k, E., \textit{On the representation of k-generalized Fibonacci and Lucas numbers}, Appl. Math. Comp. \textbf{170}(2005), 584-596.
• Shen, S., Cen, J., \textit{On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers,} Appl. Math. Comp. \textbf{216}(2010), 2891-2897.
• Solak, S., \textit{On the norms of circulant matrices with the Fibonacci and Lucas numbers}, Appl. Math. Comp. \textbf{160}(2005), 125-132.
• Solak, S., \textit{Erratum to \textquotedblleft On the Norms of Circulant Matrices with the Fibonacci and Lucas Numbers\textquotedblright \ [Appl. Math. Comp., 160, (2005), 125-132],} Appl. Math. Comp. \textbf{190}(2007), 1855-1856.
• T\"{u}rkmen, R., G\"{o}kba\c{s}, H., \textit{On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers}, Journal of Inequalities and Applications (2016),2016:65.
• Yazlik, Y., Taskara, N., \textit{On the norms of an $r-$circulant matrix with the generalized k-Horadam numbers,} Journal of Inequalities and Applications (2013), 2013:394.
Primary Language en Mathematics Articles Orcid: 0000-0002-6356-6592Author: Mustafa BAHŞİ (Primary Author)Institution: AKSARAY UNIVERSITYCountry: Turkey Orcid: 0000-0003-4085-277XAuthor: Süleyman SOLAKInstitution: Necmettin Erbakan UniversityCountry: Turkey Publication Date : December 31, 2020
 Bibtex @research article { tjmcs715052, journal = {Turkish Journal of Mathematics and Computer Science}, issn = {}, eissn = {2148-1830}, address = {}, publisher = {Matematikçiler Derneği}, year = {2020}, volume = {12}, pages = {76 - 85}, doi = {10.47000/tjmcs.715052}, title = {On The Norms of Another Form of \$r-\$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers}, key = {cite}, author = {Bahşi, Mustafa and Solak, Süleyman} } APA Bahşi, M , Solak, S . (2020). On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers . Turkish Journal of Mathematics and Computer Science , 12 (2) , 76-85 . DOI: 10.47000/tjmcs.715052 MLA Bahşi, M , Solak, S . "On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers" . Turkish Journal of Mathematics and Computer Science 12 (2020 ): 76-85 Chicago Bahşi, M , Solak, S . "On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers". Turkish Journal of Mathematics and Computer Science 12 (2020 ): 76-85 RIS TY - JOUR T1 - On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers AU - Mustafa Bahşi , Süleyman Solak Y1 - 2020 PY - 2020 N1 - doi: 10.47000/tjmcs.715052 DO - 10.47000/tjmcs.715052 T2 - Turkish Journal of Mathematics and Computer Science JF - Journal JO - JOR SP - 76 EP - 85 VL - 12 IS - 2 SN - -2148-1830 M3 - doi: 10.47000/tjmcs.715052 UR - https://doi.org/10.47000/tjmcs.715052 Y2 - 2020 ER - EndNote %0 Turkish Journal of Mathematics and Computer Science On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers %A Mustafa Bahşi , Süleyman Solak %T On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers %D 2020 %J Turkish Journal of Mathematics and Computer Science %P -2148-1830 %V 12 %N 2 %R doi: 10.47000/tjmcs.715052 %U 10.47000/tjmcs.715052 ISNAD Bahşi, Mustafa , Solak, Süleyman . "On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers". Turkish Journal of Mathematics and Computer Science 12 / 2 (December 2020): 76-85 . https://doi.org/10.47000/tjmcs.715052 AMA Bahşi M , Solak S . On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers. TJMCS. 2020; 12(2): 76-85. Vancouver Bahşi M , Solak S . On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers. Turkish Journal of Mathematics and Computer Science. 2020; 12(2): 76-85. IEEE M. Bahşi and S. Solak , "On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers", Turkish Journal of Mathematics and Computer Science, vol. 12, no. 2, pp. 76-85, Dec. 2021, doi:10.47000/tjmcs.715052

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