Determinants and Permanents of Hessenberg Matrices with Fibonacci-Like Sequences

Year 2019, Volume: 11, 65 - 71, 30.12.2019

Yasemin Taşyurdu , Fikret Işık

Abstract

In this paper, we consider Hessenberg matrices and Fibonacci-Like sequences that is defined by the recurrence relation $T_{n}=T_{n-1}+T_{n-2}$, $% n\geq 2$ and $T_{0}=m$, $T_{1}=m$ where $m$ is a fixed positive integer. We define two $n\times n$ Hessenberg matrices with applications to the Fibonacci-Like sequences and investigate the determinantal and permanental properties. We obtain that the determinants and permanents of these Hessenberg matrices are equal to the $n$th term of Fibonacci-Like sequences.

Keywords

Fibonacci sequence, Lucas sequence, Determinant, Permanent, Hessenberg matrix

References

• Brualdi, R., Gibson, P., {\em Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function}, J. Comb. Theory, \textbf{22(2)}(1977), 194--230.
• Cahill, N.D., D'Errico, J.R., Narayan, D.A., Narayan, J.Y., {\em Fibonacci determinants}, College Math. J., \textbf{33(3)}(2002), 221--225.
• H.H., {\em Permanents and determinants of tridiagonal matrices with (s, t)-Pell Lucas numbers}, International Journal of Mathematical Analysis, \textbf{11(23)}(2017), 1117--1122.
• Kayg\i s\i z, K., Sahin, A., {\em Determinant and permanent of Hessenberg matrices and Fibonacci type numbers}, Gen. Math. Notes, \textbf{9(2)}(2012), 32--41.
• K\i l\i c, E., Ta\c{s}\c{c}\i , D., {\em On the permanents of some tridiagonal matrices with applicationsto the Fibonacci and Lucas numbers}, Rocky Mt. J. Math., \textbf{37}(2007), 1953--1969.
• Koshy, T., Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.
• Koshy, T., {\em Fibonacci, Lucas and Pell numbers, and Pascal's triangle}, Math. Spectrum, \textbf{43}(2011), 125--132.
• Lee, G.Y., {\em $k$-Lucas numbers and associated bipartite graphs}, Linear Algebra Appl., \textbf{320}(2000), 51--61.
• Minc, H., Permanents, Encyclopedia Math. Appl., 6, London: Addison-Wesley Publishing Company, 1978.
• Morteza, E., {\em More on the Fibonacci sequence and Hessenberg matrices}, Integers, \textbf{6}(2006), A32.
• Ocal, A.A., Tuglu, N., Altinisik, E., {\em On the representation of k-generalized Fibonacci and Lucas numbers}, Appl. Math. Comput., \textbf{170(1)}(2005), 584--596.
• Serre, D., Matrices: Theory and Applications, New York: Springer Verlag, 2002.
• Singh, B., Sikhwal O., Bhatnagar, S., {\em Fibonacci-like sequence and its properties}, Int. J. Contemp. Math. Sciences, \textbf{5(18)}(2010), 859--868.
• Strang, G., Introduction to Linear Algebra, 2nd Ed., Wellesley MA, Wellesley-Cambridge, 1998.
• Ta\c{s}yurdu, Y., {\em On generalized Fibonacci-like sequences by Hessenberg matrices}, International Journal of Mathematics Trends and Technology, \textbf{64(1)}(2018), 51-58.
• Ta\c{s}yurdu, Y., \c{C}obanoglu N., Dilmen Z., {\em On the a new family of $k$-Fibonacci numbers}, Erzincan University Journal of Science and Technology, \textbf{9(1)}(2016), 95--101.
• Ta\c{s}yurdu, Y., G\"{u}ltekin I, {\em Determinantal and permanental representation of $q$-Fibonacci polynomials}, Qscience Connect, \textbf{25}(2014), 5p.
• Wani, A.A., Sikhwal, O.P., Sisodiya, K., {\em Relations among Fibonacci, Lucas and Fibonacci-like sequences}, International Journal of Recent Trends in Engineering \& Research, \textbf{2(9)}(2016), 125--136.
• Y\i lmaz, F., Bozkurt, D., {\em Hessenberg matrices and the Pell and Perrin numbers}, J. Number Theory., \textbf{131}(2011), 1390--1396.