Geodesic
Root location for the characteristic polynomial of a Fibonacci type sequence
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 189-195
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/CMJ.2022.0043-22
Classification : 11A63, 11B39, 11J86
Keywords: Fibonacci number; root; characteristic polynomial 
@article{10_21136_CMJ_2022_0043_22,
     author = {Du, Zhibin and da Fonseca, Carlos M.},
     title = {Root location for the characteristic polynomial of a {Fibonacci} type sequence},
     journal = {Czechoslovak Mathematical Journal},
     pages = {189--195},
     year = {2023},
     volume = {73},
     number = {1},
     doi = {10.21136/CMJ.2022.0043-22},
     mrnumber = {4541096},
     zbl = {07655762},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0043-22/}
}
TY  - JOUR
AU  - Du, Zhibin
AU  - da Fonseca, Carlos M.
TI  - Root location for the characteristic polynomial of a Fibonacci type sequence
JO  - Czechoslovak Mathematical Journal
PY  - 2023
SP  - 189
EP  - 195
VL  - 73
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0043-22/
DO  - 10.21136/CMJ.2022.0043-22
LA  - en
ID  - 10_21136_CMJ_2022_0043_22
ER  - 
%0 Journal Article
%A Du, Zhibin
%A da Fonseca, Carlos M.
%T Root location for the characteristic polynomial of a Fibonacci type sequence
%J Czechoslovak Mathematical Journal
%D 2023
%P 189-195
%V 73
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2022.0043-22/
%R 10.21136/CMJ.2022.0043-22
%G en
%F 10_21136_CMJ_2022_0043_22
Du, Zhibin; da Fonseca, Carlos M. Root location for the characteristic polynomial of a Fibonacci type sequence. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 189-195. doi: 10.21136/CMJ.2022.0043-22

[1] Bednarz, N.: On $(k,p)$-Fibonacci numbers. Mathematics 9 (2021), Article ID 727, 13 pages. | DOI

[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 | MR | JFM

[3] Glasser, M. L.: The quadratic formula made hard: A less radical approach to solving equations. Available at , 4 pages. | arXiv

[4] Janjić, M.: Determinants and recurrence sequences. J. Integer Seq. 15 (2012), Article ID 12.3.5, 21 pages. | MR | JFM

[5] Kilic, E.: The generalized order-$k$ Fibonacci-Pell sequence by matrix methods. J. Comput. Appl. Math. 209 (2007), 133-145. | DOI | MR | JFM

[6] Merca, M.: A note on the determinant of a Toeplitz-Hessenberg matrix. Spec. Matrices 1 (2013), 10-16. | DOI | MR | JFM

[7] Paja, N., Włoch, I.: Some interpretations of the $(k,p)$-Fibonacci numbers. Commentat. Math. Univ. Carol. 62 (2021), 297-307. | DOI | MR | JFM

[8] Stakhov, A., Rozin, B.: The ``golden'' algebraic equations. Chaos Solitons Fractals 27 (2006), 1415-1421. | DOI | MR | JFM

[9] Stakhov, A., Rozin, B.: Theory of Binet formulas for Fibonacci and Lucas $p$-numbers. Chaos Solitons Fractals 27 (2006), 1162-1177. | DOI | MR | JFM

[10] Trojovský, P.: On the characteristic polynomial of the generalized $k$-distance Tribonacci sequences. Mathematics 8 (2020), Article ID 1387, 8 pages. | DOI | MR

[11] Trojovský, P.: On the characteristic polynomial of $(k,p)$-Fibonacci sequence. Adv. Difference Equ. 2021 (2021), Article ID 28, 9 pages. | DOI | MR | JFM

[12] Verde-Star, L.: Polynomial sequences generated by infinite Hessenberg matrices. Spec. Matrices 5 (2017), 64-72. | DOI | MR | JFM

[13] Włoch, I.: On generalized Pell numbers and their graph representations. Commentat. Math. 48 (2008), 169-175. | MR | JFM

Cité par Sources :