A note on Q-matrices and higher order Fibonacci polynomials

Paolo Emilio Ricci
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 1, Pages 91–100
DOI: 10.7546/nntdm.2021.27.1.91-100
Full paper (PDF, 219 Kb)

Details

Authors and affiliations

Paolo Emilio Ricci
Section of Mathematics, International Telematic University UniNettuno
Corso Vittorio Emanuele II, 39, 00186, Roma, Italia

Abstract

The results described in a recent article, relative to a representation formula for the generalized Fibonacci sequences in terms of Q-matrices are extended to the case of Fibonacci, Tribonacci and R-bonacci polynomials.

Keywords

• Fibonacci numbers
• Tribonacci polynomials
• R-bonacci polynomials
• Q-matrices
• Matrix powers.

2010 Mathematics Subject Classification

• Primary 11B39
• Secondary 11B83, 12E10, 65Q30, 15A24

References

1. Beerends, R. J. (1991). Chebyshev polynomials in several variables and the radial part ofthe Laplace-Beltrami operator. Transactions of the American Mathematical Society, 328(2),779–814.
2. Brenner, J. L. (1954). Linear recurrence relations. American Mathematical Monthly, 61,171–173.
3. Bruschi, M., & Ricci, P. E. (1980). I polinomi di Lucas e di Tchebycheff in piu variabili. Rend. Mat., Ser. 6, 13, 507–530.
4. Bruschi, M., & Ricci, P. E. (1982). An explicit formula for f(A)and the generating function of the generalized Lucas polynomials. SIAM Journal on Mathematical Analysis,13, 162–165.
5. Chen, W. Y. C., & Louck, J. D. (1996). The Combinatorial Power of the Companion Matrix. Linear Algebra and its Applications, 232, 261–278.
6. Dunn, K. B., & Lidl, R. (1980). Multi-dimensional generalizations of the Chebyshev polynomials, I–II.Proceedings of the Japan Academy, 56, 154–165.
7. Gantmacher, F. R. (1959). The Theory of Matrices, Chelsea Pub. Co, New York.
8. Gould, H. W. (1981). A history of the Fibonacci Q-matrix and a higher-dimensional problem.The Fibonacci Quarterly, 19, 250–257.
9. Hirst, H. P., & Macey, W. T. (1997). Bounding the Roots of Polynomials, The College Mathematics Journal, 28(4), 292–295
10. Hoggatt, V. E., & Bicknell, M. (1973). Generalized Fibonacci polynomials. The Fibonacci Quarterly, 11, 457–465.
11. Ivie, J. (1972). A general Q-matrix,The Fibonacci Quarterly, 10(3), 255–261, 264.
12. Koornwinder, T. H. (1974). Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, I–II.Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Ser. A, 77, 46–66.
13. Koornwinder, T. H. (1974). Orthogonal polynomials in two variables which areeigen functions of two algebraically independent partial differential operators, III–IV.Indagationes Mathematicae, 36, 357–381.
14. Lidl, R. (1975). Tschebyscheffpolynome in mehreren variabelen. Journal fur die reine undangewandte Mathematik, 273, 178–198.
15. Lidl, R., & Wells, C. (1972). Chebyshev polynomials in several variables. Journal fur diereine und angewandte Mathematik, 255, 104–111.
16. Lucas, E. (1891).Theorie des Nombres. Gauthier-Villars, Paris.
17. Miles Jr., E.P. (1960). Generalized Fibonacci numbers and associated matrices. The American Mathematical Monthly, 67, 745–752.
18. Parodi, M. (1959). La Localisation des Valeurs Caract eristiques des Matrices et ses Applications. Gauthier-Villars, Paris.
19. Raghavacharyulu, I. V. V., & Tekumalla, A. R. (1972). Solution of the Difference Equations of Generalized Lucas Polynomials. Journal of Mathematical Physics, 13, 321–324.
20. Ricci, P. E. (1976). Sulle potenze di una matrice. Rend. Mat., Ser. 6, 9, 179–194.
21. Ricci, P. E. (2020). A note on Golden ratio and higher order Fibonacci sequences. Turkish Journal of Analysis and Number Theory, 8(1), 1–5.
22. Singh, P. (1985). The so-called Fibonacci numbers in ancient and medieval India. Historia Mathematica, 12, 229–244.
23. Rosenbaum, R. A. (1959). An application of matrices to linear recursion relations. The American Mathematical Monthly, 66, 792–793.
24. Yordzhev, K. (2014). Factor-set of binary matrices and Fibonacci numbers. Applied Mathematics and Computation, 236, 235–238