A note on certain matrices with h(x)-Fibonacci polynomials
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Abstract
In this paper it is considered a g-circulant, right circulant, left circulant and a special kind of a tridiagonal matrices whose entries are h(x)-Fibonacci polynomials. The determinant of these matrices is established and with the tridiagonal matrices we show that the determinant is equal to the $n$th term of the h(x)-Fibonacci polynomials.
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How to Cite
Catarino, P.
(2017).
A note on certain matrices with h(x)-Fibonacci polynomials.
Acta Mathematica Universitatis Comenianae, 86(2), 263-270.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/386/469
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References
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Catarino P., Vasco P., Borges A., Campos H., Aires A. P., Sums, Products and identities involving k-Fibonacci and k-Lucas sequences, JP J. Algebra Number Theory Appl. 32 (1) (2014), 63-77.
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Falcon S., On the generating matrices of the k-Fibonacci numbers, Proyecciones Journal of Mathematics 32 (4) (2013), 347-357.
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Good I. J., On the inversion of circulant matrices, Biometrika 37 (1950), 185-186.
Ipek A., On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries, Applied Mathematics and Computation 217 (2011), 6011-6012.
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KiliÂc E., Tasci D. and Haukkanen P., On the generalized Lucas sequences by Hessenberg matrices, Ars Combinatoria 95 (2010), 383-395.
Nalli A. and Haukkanen P., On generalized Fibonacci and Lucas polynomials, Chaos, Solitons and Fractals 42 (2009), 3179-3186.
Qu W., Lei S. L. and Vong S. W., Circulant and skew-circulant splitting iteration for fractional advection-diffusion equations, International Journal of Computer Mathematics, 2014.
http://dx.doi.org/10.1080/00207160.2013.871001
Shen S. Q., Cen J. M. and Hao Y., On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers, Applied Mathematics and Computation 217 (2011), 9790-9797.
Solak S., On the norms of circulant matrices with the Fibonacci and Lucas numbers, Applied Mathematics and Computation 160 (2005), 125-132.
Yazlik Y. and Taskara N., Spectral norm, Eigenvalues and Determinant of Circulant Matrix involving the Generalized k-Horadam numbers, Ars Combinatoria 104 (2012), 505-512.
Yazlik Y. and Taskara N., On the Inverse of Circulant Matrix via Generalized k-Horadam Numbers, Applied Mathematics and Computation 223 (2013), 191-196.
Zhou J. and Jiang Z., The spectral norms of g-circulant matrices with classical Fibonacci and Lucas numbers entries, Applied Mathematics and Computation 233 (2014), 582-587.
Wu J. and Zou X., Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, Journal of Differential Equations 135 (2) (1997), 315-357.
Wu Y. K., Jia R. Z. and Li Q., g-circulant solutions to the (0; 1) matrix equation A^m = J^*_n, Linear Algebra Appl. 345 (2002), 195-224.
Bolat C. and Kose H., On the Properties of k-Fibonacci Numbers, Int. J. Contemp. Math. Sci. 22 (5) (2010), 1097-1105.
Bozkurt D. and Tam T., Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers, Applied Mathematics and Computations 219 (2012), 544-551.
Catarino P., On some identities for k-Fibonacci sequence, Int. J. Contemp. Math. Sci. 9 (1) (2014), 37-42.
Catarino P., Vasco P., Borges A., Campos H., Aires A. P., Sums, Products and identities involving k-Fibonacci and k-Lucas sequences, JP J. Algebra Number Theory Appl. 32 (1) (2014), 63-77.
Davis P. J., Circulant Matrices, John Wiley & Sons, New York, 1979.
Erbas C. and Tanik M. M., Generating solutions to the N-Queens problems using 2-circulants}, Math. Mag. 68 (1995), 343-356.
Falcon S., On the generating matrices of the k-Fibonacci numbers, Proyecciones Journal of Mathematics 32 (4) (2013), 347-357.
Gong Y., Jiang Z. L. and Gao Y., On Jacobsthal and Jacobsthal-Lucas Circulant Type Matrices, Abstract and Applied Analysis, (2014), Article ID 418293, 11 pages.
Good I. J., On the inversion of circulant matrices, Biometrika 37 (1950), 185-186.
Ipek A., On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries, Applied Mathematics and Computation 217 (2011), 6011-6012.
Jiang Z. L. and Zhou Z. X., Circulant Matrices, Chengdu Technology University Publishing Company, Chengdu, 1999.
Jiang Z. L., Gong Y. and Gao Y., Circulant type matrices with the sum and product of Fibonacci and Lucas numbers, Abstract and Applied Analysis, vol. 2014, Article ID 375251, 12 pages, 2014.
KiliÂc E., Tasci D. and Haukkanen P., On the generalized Lucas sequences by Hessenberg matrices, Ars Combinatoria 95 (2010), 383-395.
Nalli A. and Haukkanen P., On generalized Fibonacci and Lucas polynomials, Chaos, Solitons and Fractals 42 (2009), 3179-3186.
Qu W., Lei S. L. and Vong S. W., Circulant and skew-circulant splitting iteration for fractional advection-diffusion equations, International Journal of Computer Mathematics, 2014.
http://dx.doi.org/10.1080/00207160.2013.871001
Shen S. Q., Cen J. M. and Hao Y., On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers, Applied Mathematics and Computation 217 (2011), 9790-9797.
Solak S., On the norms of circulant matrices with the Fibonacci and Lucas numbers, Applied Mathematics and Computation 160 (2005), 125-132.
Yazlik Y. and Taskara N., Spectral norm, Eigenvalues and Determinant of Circulant Matrix involving the Generalized k-Horadam numbers, Ars Combinatoria 104 (2012), 505-512.
Yazlik Y. and Taskara N., On the Inverse of Circulant Matrix via Generalized k-Horadam Numbers, Applied Mathematics and Computation 223 (2013), 191-196.
Zhou J. and Jiang Z., The spectral norms of g-circulant matrices with classical Fibonacci and Lucas numbers entries, Applied Mathematics and Computation 233 (2014), 582-587.
Wu J. and Zou X., Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, Journal of Differential Equations 135 (2) (1997), 315-357.
Wu Y. K., Jia R. Z. and Li Q., g-circulant solutions to the (0; 1) matrix equation A^m = J^*_n, Linear Algebra Appl. 345 (2002), 195-224.