Root location for the characteristic polynomial of a Fibonacci type sequence

Zhibin Du; Carlos Martins da Fonseca

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 1, page 189-195
  • ISSN: 0011-4642

Abstract

top
We analyse the roots of the polynomial x n - p x n - 1 - q x - 1 for p ⩾ q ⩾ 1 . This is the characteristic polynomial of the recurrence relation F k , p , q ( n ) = p F k , p , q ( n - 1 ) + q F k , p , q ( n - k + 1 ) + F k , p , q ( n - k ) for n ⩾ k , which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.

How to cite

top

Du, Zhibin, and da Fonseca, Carlos Martins. "Root location for the characteristic polynomial of a Fibonacci type sequence." Czechoslovak Mathematical Journal 73.1 (2023): 189-195. <http://eudml.org/doc/299592>.

@article{Du2023,
abstract = {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 $F_\{k,p,q\}(n) = pF_\{k,p,q\}(n- 1) + qF_\{k,p,q\}(n-k + 1) + F_\{k,p,q\}(n-k)$ 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.},
author = {Du, Zhibin, da Fonseca, Carlos Martins},
journal = {Czechoslovak Mathematical Journal},
keywords = {Fibonacci number; root; characteristic polynomial},
language = {eng},
number = {1},
pages = {189-195},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Root location for the characteristic polynomial of a Fibonacci type sequence},
url = {http://eudml.org/doc/299592},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Du, Zhibin
AU - da Fonseca, Carlos Martins
TI - Root location for the characteristic polynomial of a Fibonacci type sequence
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 1
SP - 189
EP - 195
AB - 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 $F_{k,p,q}(n) = pF_{k,p,q}(n- 1) + qF_{k,p,q}(n-k + 1) + F_{k,p,q}(n-k)$ 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.
LA - eng
KW - Fibonacci number; root; characteristic polynomial
UR - http://eudml.org/doc/299592
ER -

References

top
  1. Bednarz, N., 10.3390/math9070727, Mathematics 9 (2021), Article ID 727, 13 pages. (2021) DOI10.3390/math9070727
  2. Fonseca, C. M. da, 10.1007/s10587-011-0059-1, Czech. Math. J. 61 (2011), 917-921. (2011) Zbl1249.15011MR2886247DOI10.1007/s10587-011-0059-1
  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. (1994) 
  4. Janjić, M., Determinants and recurrence sequences, J. Integer Seq. 15 (2012), Article ID 12.3.5, 21 pages. (2012) Zbl1286.11017MR2908736
  5. Kilic, E., 10.1016/j.cam.2006.10.071, J. Comput. Appl. Math. 209 (2007), 133-145. (2007) Zbl1162.11013MR2387120DOI10.1016/j.cam.2006.10.071
  6. Merca, M., 10.2478/spma-2013-0003, Spec. Matrices 1 (2013), 10-16. (2013) Zbl1291.15015MR3155395DOI10.2478/spma-2013-0003
  7. Paja, N., Włoch, I., 10.14712/1213-7243.2021.026, Commentat. Math. Univ. Carol. 62 (2021), 297-307. (2021) Zbl07442493MR4331284DOI10.14712/1213-7243.2021.026
  8. Stakhov, A., Rozin, B., 10.1016/j.chaos.2005.04.107, Chaos Solitons Fractals 27 (2006), 1415-1421. (2006) Zbl1148.11009MR2164865DOI10.1016/j.chaos.2005.04.107
  9. Stakhov, A., Rozin, B., 10.1016/j.chaos.2005.04.106, Chaos Solitons Fractals 27 (2006), 1162-1177. (2006) Zbl1178.11018MR2164849DOI10.1016/j.chaos.2005.04.106
  10. Trojovský, P., 10.3390/math8081387, Mathematics 8 (2020), Article ID 1387, 8 pages. (2020) MR4197344DOI10.3390/math8081387
  11. Trojovský, P., 10.1186/s13662-020-03186-8, Adv. Difference Equ. 2021 (2021), Article ID 28, 9 pages. (2021) Zbl1485.11038MR4197344DOI10.1186/s13662-020-03186-8
  12. Verde-Star, L., 10.1515/spma-2017-0002, Spec. Matrices 5 (2017), 64-72. (2017) Zbl1360.15034MR3602625DOI10.1515/spma-2017-0002
  13. Włoch, I., On generalized Pell numbers and their graph representations, Commentat. Math. 48 (2008), 169-175. (2008) Zbl1175.05105MR2482763

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.