On the connection between tridiagonal matrices, Chebyshev polynomials, and Fibonacci numbers

Carlos M. da Fonseca

doi:10.2478/ausm-2020-0019

Abstract

In this note, we recall several connections between the determinant of some tridiagonal matrices and the orthogonal polynomials allowing the relation between Chebyshev polynomials of second kind and Fibonacci numbers. With basic transformations, we are able to recover some recent results on this matter, bringing them into one place.

References

[1] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ninth edition, Dover Publications, Inc., New York, 1970.
Search in Google Scholar Back to article
[2] M. Andelić, Z. Du, C. M. da Fonseca, E. Kılıç, A matrix approach to some second-order difference equations with sign-alternating coefficients, J. Difference Equ. Appl., 26 (2020), 149–162.10.1080/10236198.2019.1709180
Search in Google Scholar Back to article
[3] M. Anđelić, C. M. da Fonseca, A determinantal formula for generalized Fibonacci numbers, Matematiche (Catania), 74 (2019), 363–367.
Search in Google Scholar Back to article
[4] G. B. Arfken, H. J. Weber, F. E. Harris, Mathematical Methods for Physicists: A Comprehensive Guide, Academic Press, Orlando, 1985.
Search in Google Scholar Back to article
[5] R. G. Buschman, Fibonacci numbers, Chebyshev polynomials generalizations and difference equations, Fibonacci Quart. 1 (1963), 1–8, 19.
Search in Google Scholar Back to article
[6] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
Search in Google Scholar Back to article
[7] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht, Netherlands, 1974.
Search in Google Scholar Back to article
[8] Erdélyi, et al., Higher Transcendental Functions, vol. 2, McGraw Hill, New York, 1953.
Search in Google Scholar Back to article
[9] C. F. Fischer, R. A. Usmani, Properties of some tridiagonal matrices and their application to boundary value problems, SIAM J. Numer. Anal., 6 (1969), 127–141.10.1137/0706014
Search in Google Scholar Back to article
[10] C. M. da Fonseca, J. Petronilho, Explicit inverses of some tridiagonal matrices, Linear Algebra Appl., 325 (2001), 7–21.10.1016/S0024-3795(00)00289-5
Search in Google Scholar Back to article
[11] G. Heinig, K. Rost, Algebraic Methods for Toeplitz-Like Matrices and Operators, Birkhäuser OT 13, 1984.10.1515/9783112529003
Search in Google Scholar Back to article
[12] A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart., 3 (1965), 161–176.
Search in Google Scholar Back to article
[13] F. Qi, V. Čerňanová, Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 81 (2019), 123–136.
Search in Google Scholar Back to article
[14] F. Qi, V. Čerňanová, X.-T. Shi, B.-N. Guo, Some properties of central Delannoy numbers, J. Comput. Appl. Math., 328 (2018), 101–115.10.1016/j.cam.2017.07.013
Search in Google Scholar Back to article
[15] F. Qi, B.-N. Guo, Expressing the generalized Fibonacci polynomials in terms of a tridiagonal determinant, Matematiche (Catania), 72 (2017), 167–175.
Search in Google Scholar Back to article
[16] F. Qi, A.-Q. Liu, Alternative proofs of some formulas for two tridiagonal determinants, Acta Univ. Sapientiae, Mathematica, 10 (2018), 287–297.
Search in Google Scholar Back to article
[17] G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl., 13 (1992), 707–728.10.1137/0613045
Search in Google Scholar Back to article
[18] D. Moskovitz, The numerical solution of Laplace’s and Poisson’s equations, Quart. Appl. Math., 2 (1944), 14S–63.10.1090/qam/11186
Search in Google Scholar Back to article
[19] P. Schlegel, The explicit inverse of a tridiagonal matrix, Math. Comp. 24 (1970), 665.10.1090/S0025-5718-1970-0273798-2
Search in Google Scholar Back to article
[20] R. Usmani, Inversion of a tridiagonal Jabobi matrix, Linear Algebra Appl., 212/213 (1994), 413–414.10.1016/0024-3795(94)90414-6
Search in Google Scholar Back to article
[21] T. Yamamoto, Y. Ikebe, Inversion of band matrices, Linear Algebra Appl., 24 (1979), 105–111.10.1016/0024-3795(79)90151-4
Search in Google Scholar Back to article

On the connection between tridiagonal matrices, Chebyshev polynomials, and Fibonacci numbers

Abstract

Paradigm

My account