Spectral properties of a functional binomial matrix

Year 2024, Volume: 73 Issue: 3, 749 - 764, 27.09.2024

Morteza Bayat

https://doi.org/10.31801/cfsuasmas.1360864

Abstract

In this article, we consider the definition of the Fibonacci polynomial sequence with the second-order linear recurrence relation, where coefficients and initial conditions depend on the variable $t$. And then, we introduce the functional binomial matrix depending on the coefficients of the second-order linear recurrence relation. In the following, we study the spectral properties of the functional binomial matrix using the Fibonacci polynomial sequence and we obtain a diagonal decomposition for it using the Vandermunde matrix. Finally, by applying some linear algebra tools we obtain a number of combinatorial identities involving the Fibonacci polynomial sequence.

Keywords

Functional Fibonacci matrix, generalized Fibonacci sequence, generalized Fibonacci polynomial, characteristic polynomial, Pascal matrix, functional binomial matrix

References

• Akkuse, I., The eigenvectors of a combinatorial matrix, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 60 (2011), 9—14.
• Bayat, M., Teimoori, H., The linear algebra of the generalized Pascal functional matrix, Linear Algebra and its Application, 295 (1999), 81-89. https://doi.org/10.1016/S0024-3795(99)00062-2
• Bayat, M. Generalized Pascal k-eliminted functional matrix with $2n$ variables, Electronic Journal of Linear Algebra, 22 (2010), 419-429. https://doi.org/10.13001/1081-3810.1446
• Berg, C., Fibonacci numbers and orthogonal polynomials, Arab Journal of Mathematical Sciences, 17 (2011), 75-88. https://doi.org/10.48550/arXiv.math/0609283
• Bergum, G. E., Hoggatt, V. E. Jr., Irreducibility of Lucas and generalized Lucas polynomials, Fibonacci Quart., 12 (1974), 95–100.
• Call, G. S., Velleman, D. J., Pascal’s matrices, Amer. Math. Monthly, 100 (1993), 372-376. https://doi.org/10.1080/00029890.1993.11990415
• Carlitz, L., The Characteristic polynomial of a certain matix of binomial coefficients, Fibonacci Quarterly, 3 (1965), 81-89.
• Catalan, E. C., Notes surla theorie des fractions continuess et sur certaines series, Mem. Acad. R. Belgique, 45 (1883), 1-82.
• Florez, R., McAnally, N., Mukherjee, A., Identities for the generalized Fibonacci polynomial, http://arxiv.org/abs/1702.01855v2.
• Gupta, V. K., Panwar, Y. K., Sikhwal, O., Generalized Fibonacci sequences, Theoretical Mathematics & Applications, 2 (2012), 115-124.
• Horadam, A. F., Jacobsthal representation numbers, The Fib. Quart., 34 (1996), 40-54.
• Horadam, A. F., Mahon, J. M., Pell and Pell-Lucas polynomials, Fib. Quart., 23 (1985), 7-20.
• Jacobsthal, E., Fibonacci polynome und kreisteil ungsgleichugen sitzungsberichte der Berliner, Math. Gesellschaft, 17 (1919-20), 43-57.
• Kalman D., Mena, R., The Fibonacci numbers: exposed, The Mathematical Magazine, 76 (2003), 167-181. https://doi.org/10.2307/3219318
• Kaygisiz, K., Sahin, A., New generalizations of Lucas numbers, Gen. Math. Notes, 10 (2012), 63-77.
• Kizilaslan, G., The linear algebra of a generalized Tribonacci matrix, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 72 (2023), 169–181. https://doi.org/10.31801/cfsuasmas.1052686
• Koshy, T., Fibonacci and Lucas Numbers with Applications, Toronto, New York, NY, USA, 2001.
• Lee, G. Y., Asci, M., Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials, Journal of Applied Mathematics, (2012), ArticleID 264842, 18 pages. https://doi.org/10.1155/2012/264842
• Lupas, A., A guide of Fibonacci and Lucas polynomial, Octagon Mathematics Magazine, 7 (1999), 2-12.
• Mericier, A., Identities containing Gauss binomial coefficients, Discrate Math., 76 (1989), 67-73. https://doi.org/10.1016/0012-365X(89)90290-2
• Nalli, A., Haukkanen, P., On generalized Fibonacci and Lucas polynomials, Chaos, Solitons and Fractals, 42 (2009), 3179–3186. https://doi.org/10.1016/j.chaos.2009.04.048
• Panwar, Y. K., Singh, B., Gupta, V. K., Generalized Fibonacci polynomials, Turkish Journal of Analysis and Number Theory, 1 (2013), 43-47.
• Postavaru, O., An efficient numerical method based on Fibonacci polynomials to solve fractional differential equations, Mathematics and Computers in Simulation, 212 (2023), 406-422. https://doi.org/10.1016/j.matcom.2023.04.028
• Prasolov, V. V., Problems and Theorems in Linear Algebra, American Mathematical Society, 1994.
• Sikhwal, O., Vyas, Y., Generalized Fibonacci polynomials and some fundamental properties, Scirea Journal of Mathematics, 1 (2016), 16-23.
• Singh Sikhwal, B. O., Bhatnagar, S., Fibonacci-like sequence and its properties, Int. J. Contemp. Math. Sciences, 5 (2010), 859-868.
• Stanimirovic, P., Nikolov, J., Stanimirovic, I., A generalization of Fibonacci and Lucas matrices, Discrete Applied Mathematics, 156 (2008), 2606–2619. https://doi.org/10.1016/j.dam.2007.09.028