Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
Fibonacci number; root; characteristic polynomial
Fibonacci number; root; characteristic polynomial
Summary:
We analyse the roots of the polynomial $x^n-px^{n-1}-qx-1$ for $p\geqslant q\geqslant 1$. This is the characteristic polynomial of the recurrence relation $F_{k,p,q}(n) = pF_{k,p,q}(n- \nobreak 1) + qF_{k,p,q}(n-k + 1) + F_{k,p,q}(n-k)$ for $n \geqslant k$, which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.
References:
[1] Bednarz, N.: On $(k,p)$-Fibonacci numbers. Mathematics 9 (2021), Article ID 727, 13 pages. DOI 10.3390/math9070727
[2] Fonseca, C. M. da: An identity between the determinant and the permanent of Hessenberg-type matrices. Czech. Math. J. 61 (2011), 917-921. DOI 10.1007/s10587-011-0059-1 | MR 2886247 | Zbl 1249.15011
[3] Glasser, M. L.: The quadratic formula made hard: A less radical approach to solving equations. Available at https://arxiv.org/abs/math/9411224 (1994), 4 pages.
[4] Janjić, M.: Determinants and recurrence sequences. J. Integer Seq. 15 (2012), Article ID 12.3.5, 21 pages. MR 2908736 | Zbl 1286.11017
[5] Kilic, E.: The generalized order-$k$ Fibonacci-Pell sequence by matrix methods. J. Comput. Appl. Math. 209 (2007), 133-145. DOI 10.1016/j.cam.2006.10.071 | MR 2387120 | Zbl 1162.11013
[6] Merca, M.: A note on the determinant of a Toeplitz-Hessenberg matrix. Spec. Matrices 1 (2013), 10-16. DOI 10.2478/spma-2013-0003 | MR 3155395 | Zbl 1291.15015
[7] Paja, N., Włoch, I.: Some interpretations of the $(k,p)$-Fibonacci numbers. Commentat. Math. Univ. Carol. 62 (2021), 297-307. DOI 10.14712/1213-7243.2021.026 | MR 4331284 | Zbl 07442493
[8] Stakhov, A., Rozin, B.: The ``golden'' algebraic equations. Chaos Solitons Fractals 27 (2006), 1415-1421. DOI 10.1016/j.chaos.2005.04.107 | MR 2164865 | Zbl 1148.11009
[9] Stakhov, A., Rozin, B.: Theory of Binet formulas for Fibonacci and Lucas $p$-numbers. Chaos Solitons Fractals 27 (2006), 1162-1177. DOI 10.1016/j.chaos.2005.04.106 | MR 2164849 | Zbl 1178.11018
[10] Trojovský, P.: On the characteristic polynomial of the generalized $k$-distance Tribonacci sequences. Mathematics 8 (2020), Article ID 1387, 8 pages. DOI 10.3390/math8081387 | MR 4197344
[11] Trojovský, P.: On the characteristic polynomial of $(k,p)$-Fibonacci sequence. Adv. Difference Equ. 2021 (2021), Article ID 28, 9 pages. DOI 10.1186/s13662-020-03186-8 | MR 4197344 | Zbl 1485.11038
[12] Verde-Star, L.: Polynomial sequences generated by infinite Hessenberg matrices. Spec. Matrices 5 (2017), 64-72. DOI 10.1515/spma-2017-0002 | MR 3602625 | Zbl 1360.15034
[13] Włoch, I.: On generalized Pell numbers and their graph representations. Commentat. Math. 48 (2008), 169-175. MR 2482763 | Zbl 1175.05105
Search
Partner of
