The Relation Between Chebyshev Polynomials and Jacobsthal and Jacobsthal Lucas Sequences

Year 2020, Volume: 24 Issue: 6, 1162 - 1170, 01.12.2020

Şükran Uygun

https://doi.org/10.16984/saufenbilder.687708

Abstract

In this paper Jacobsthal, Jacobsthal Lucas and generalized Jacobsthal sequences are denoted by aid of first or second type of Chebyshev polynomials by different equalities. Then using these equalities a relation is obtained between Jacobsthal and generalized Jacobsthal numbers. Moreever, the nth powers of some special matrices are found by using Jacobsthal numbers or Chebyshev polynomials. Some connections among Jacobsthal, Jacobsthal Lucas are revealed by using the determinant of the power of some special matrices. Then, the properties of Jacobsthal, Jacobsthal Lucas numbers are obtained by using the identities of Chebyshev polynomials.

Keywords

Chebhshev polynomials, Jacobsthal Sequence, Jacobsthal Lucas Sequence

References

• A. F. Horadam, “Basic Properties of a certain generalized Squence of Numbers”, Fibonacci Quarterly, pp. 161-176, 1965.
• A. F. Horadam, “Special Properties of the Sequence {Wn(a,b;p,q)}”, Fibonacci Quarterly, vol. 5, pp. 424-434, 1967.
• A. F. Horadam, “Tschebyscheff and Other Functions Associated with the Sequence”, Fibonacci Quarterly, vol. 7, no. 1, pp. 14-22, 1969.
• A. F. Horadam, “Jacobsthal representation numbers”, The Fibonacci Quarterly, vol. 37, no. 2, pp. 40-54, 1996.
• T. Koshy, “Fibonacci and Lucas Numbers with Applications”, John Wiley and Sons Inc., NY 2001.
• G. Udrea, “A note on Sequence of A. F. Horadam,” Portugalia Mathematica, vol. 53, no. 24, pp. 143-144, 1996.
• T. Mansour, “A formula for the generating functions of powers of Horadam sequence”, Australasian Journal of Combinatorics, vol. 30, pp, 207-212, 2004.
• T. Horzum and E. G. Kocer, “On Some Properties of Horadam Polynomials”, Int math. Forum, vol. 4, no. 25-28, pp. 1243-1252, 2009.
• E. Kilic and E Tan, “On Binomial Sums for the General Second Order Linear Recurrence”, Integers Electronic Journal of Combnatorial Number Theory, vol. 10, pp. 801-806, 2010.
• N. Taskara, K.Uslu, Y. Yazlık and N. Yılmaz “The Construction of Horadam Numbers in Terms of the Determinant of Tridioganal Matrices”, Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, vol. 1389, pp. 367-370, 2011.
• C. K. Ho and C. Y. Chong, “Odd and even sums of generalized Fibonacci numbers by matrix methods”. Am. Inst. Phys. Conf. Ser., vol. 1602, pp. 1026-1032, 2014.
• S. P. Jun and K. H. Choi, “Some properties of the Generalized Fibonacci Sequence by Matrix Methods”, Korean J. Math, vol. 24, no. 4, pp. 681-691, 2016.
• S. Uygun, “The (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas sequences”, Applied Mathematical Sciences, vol. 9, no. 7, pp. 3467-3476, 2015.
• S. Uygun, “The Combinatorial Representation of Jacobsthal and Jacobsthal Lucas Matrix Sequences”, Ars Combinatoria, vol. 135, pp. 83-92, 2017.
• S. Uygun, “A New Generalization for Jacobsthal and Jacobsthal Lucas Sequences”, Asian Journal of Mathematics and Physics, vol. 2, no. 1, pp. 14-21, 2018.
• G. Udrea, “A Problem of Diophantos-Fermat and Chebyshev polynomials of the second kind”, Portugalia Mathematica, vol. 52, pp. 301-304, 1995.