On a sequence derived from the Laplace transform of the characteristic polynomial of the Fibonacci sequence

Carlos M. da Fonseca and Anthony G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 3, Pages 557–563
DOI: 10.7546/nntdm.2023.29.3.557-563
Full paper (PDF, 350 Kb)

Details

Authors and affiliations

Carlos M. da Fonseca
Kuwait College of Science and Technology,
Doha District, Safat 13133, Kuwait
Chair of Computational Mathematics, University of Deusto
48007 Bilbao, Spain

Anthony G. Shannon
Honorary Fellow, Warrane College, University of New South Wales, 2033, Australia

Abstract

Recently, based on the Laplace transform of the characteristic polynomial of the Fibonacci sequence, Deveci and Shannon established a new sequence and analysed some of its properties. They disclosed in particular the odd terms. In this short note, we provide a matricial representation for this sequence as well as one in terms of the Chebyshev polynomials of the second kind. The subsequence of the even terms is also disclosed.

Keywords

• Fibonacci sequence
• Recurrence
• Chebyshev polynomials of the second kind
• Determinant
• Tridiagonal matrices

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 15A15

References

1. Aharonov, D., Beardon, A., & Driver, K. (2005). Fibonacci, Chebyshev, and orthogonal polynomials. The American Mathematical Monthly, 112, 612–630.
2. Anđelić, M., Du, Z., da Fonseca, C. M., & Kılıç, E. (2020). Second-order difference equations with sign-alternating coefficients. Journal of Difference Equations and Applications, 26, 149–162.
3. Anđelić, M., & da Fonseca, C. M. (2021). Determinantal representations for the number of subsequences without isolated odd terms. Notes on Number Theory and Discrete Mathematics, 27(4), 116–121.
4. Anđelić, M., & da Fonseca, C. M. (2021). On the constant coefficients of a certain recurrence relation: A simple proof. Heliyon, 7, Article ID E07764.
5. Anđelić, M., da Fonseca, C. M., & Yılmaz, F. (2022). The bi-periodic Horadam sequence and some perturbed tridiagonal 2-Toeplitz matrices: A unified approach. Heliyon, 8, Article ID E08863.
6. Buschman, R. G. (1963). Fibonacci numbers, Chebyshev polynomials generalizations and difference equations. The Fibonacci Quarterly, 1, 1–8, 19.
7. Cvetković, A., Rajković, P., & Ivković, M. (2002). Catalan numbers, the Hankel transform, and Fibonacci numbers. Journal of Integer Sequences, 5, Article ID 02.1.3.
8. Deveci, Ö., & Shannon, A. G. (2022). On recurrence results from matrix transforms. Notes on Number Theory and Discrete Mathematics, 28(4), 589–592.
9. Du, Z., Dimitrov, D., & da Fonseca, C. M. (2021). New strong divisibility sequences. Ars Mathematica Contemporanea, 22(1), Article ID P1.08.
10. Dumont, D. (1974). Interprétations combinatoires des nombres de Genocchi. Duke Mathematical Journal, 41, 305–318.
11. Estrada, E., & de la Peña, J. A. (2013). Integer sequences from walks in graphs. Notes on Number Theory and Discrete Mathematics, 19(3), 78–84.
12. Flegg, G. (1983). Numbers: Their History and Meaning. Schocken Books, New York.
13. Da Fonseca, C. M. (2020). On the connection between tridiagonal matrices, Chebyshev polynomials, and Fibonacci numbers. Acta Universitatis Sapientiae Mathematica, 12, 280–286.
14. Da Fonseca, C. M. (2014). Unifying some Pell and Fibonacci identities. Applied Mathematics and Computation, 236, 41–42.
15. Gould, H. W. (1972). Combinatorial Identities. Morgantown: Morgantown Printing and Binding Co.
16. Hong, S., & Kwak, J. H. (1993). Regular fourfold coverings with respect to the identity automorphism. Journal of Graph Theory, 17, 621–627.
17. Horadam, A. F. (1965). Generating functions for powers of a certain generalised sequence of numbers. Duke Mathematical Journal, 32(3), 437–446.
18. Horadam, A. F. (1969). Tschebyscheff and other functions associated with the sequence . The Fibonacci Quarterly, 7, 14–22.
19. Horadam, A. F., & Shannon, A. G. (1987). Generalization of identities of Catalan and others. Portugaliae Mathematica, 44(2), 137–148.
20. Melham, R. S., & Shannon, A. G. (1995). On reciprocal sums of Chebyshev related
    sequences. The Fibonacci Quarterly, 33(2), 194–202.
21. Shannon, A. G. (1975). Fibonacci analogs of the classical polynomials. Mathematics Magazine, 48(3), 123–130.
22. Singh, P. (1985). The so-called Fibonacci numbers in ancient and medieval India. Historia Mathmatica, 12(3), 229–244.
23. Thiele, T. N. (1909). Interpolationsrechnung. Leipzig: B. G. Teubner.
24. Udrea, G. (1996). A note on the sequence of A. F. Horadam. Portugaliae
    Mathematica, 53(2), 143–155.

Manuscript history

• Received: 11 February 2023
• Revised: 18 July 2023
• Accepted: 29 July 2023
• Online First: 30 July 2023

Copyright information

Ⓒ 2023 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).