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Symmetry
Volume 11
Issue 6
10.3390/sym11060782
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Article

A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant

by
Feng Qi
1,2,3,
Can Kızılateş
4 and
Wei-Shih Du
5,*
1
Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, China
2
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China
3
School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China
4
Department of Mathematics, Faculty of Arts and Sciences, Zonguldak Bülent Ecevit University, Zonguldak 67100, Turkey
5
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(6), 782; https://doi.org/10.3390/sym11060782
Submission received: 19 May 2019 / Revised: 8 June 2019 / Accepted: 11 June 2019 / Published: 12 June 2019
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
Versions Notes

Abstract

In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants.
Keywords: closed formula; Horadam polynomial, tridiagonal determinant; generalized Fibonacci polynomial; Lucas polynomial; Pell–Lucas polynomial; Chebyshev polynomial of the first kind; Chebyshev polynomial of the second kind closed formula; Horadam polynomial, tridiagonal determinant; generalized Fibonacci polynomial; Lucas polynomial; Pell–Lucas polynomial; Chebyshev polynomial of the first kind; Chebyshev polynomial of the second kind

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MDPI and ACS Style

Qi, F.; Kızılateş, C.; Du, W.-S. A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant. Symmetry 2019, 11, 782. https://doi.org/10.3390/sym11060782

AMA Style

Qi F, Kızılateş C, Du W-S. A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant. Symmetry. 2019; 11(6):782. https://doi.org/10.3390/sym11060782

Chicago/Turabian Style

Qi, Feng, Can Kızılateş, and Wei-Shih Du. 2019. "A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant" Symmetry 11, no. 6: 782. https://doi.org/10.3390/sym11060782

APA Style

Qi, F., Kızılateş, C., & Du, W.-S. (2019). A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant. Symmetry, 11(6), 782. https://doi.org/10.3390/sym11060782

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