Some identities involving Chebyshev polynomials, Fibonacci polynomials and their derivatives

Jugal Kishore and Vipin Verma
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 2, Pages 204–215
DOI: 10.7546/nntdm.2023.29.2.204-215
Full paper (PDF, 186 Kb)

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Authors and affiliations

Jugal Kishore
Department of Mathematics, School of Chemical Engineering and Physical Sciences,
Lovely Professional University, Phagwara 144411, Punjab, India

Vipin Verma
SVKM’s Narsee Monjee Institute of Management Studies (NMIMS) University
V.L. Mehta Road, Vile Parle (West) Mumbai, Maharashtra 400056, India

Abstract

In this paper, we will derive the explicit formulae for Chebyshev polynomials of the third and fourth kind with odd and even indices using the combinatorial method. Similar results are also deduced for their r-th derivatives. Finally, some identities involving Chebyshev polynomials of the third and fourth kind with even and odd indices and Fibonacci polynomials with negative indices are obtained.

Keywords

• Chebyshev polynomials
• Fibonacci polynomials
• Orthogonality

2020 Mathematics Subject Classification

• 11B39
• 11B83
• 33C47

References

1. Abramowitz, M., & Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.
2. Doman, B. G. S. (2015). The Classical Orthogonal Polynomials. World Scientific Publishing Company, Singapore.
3. Falcón, S., & Plaza, A. (2007). On the Fibonacci k-numbers. Chaos, Solitons & Fractals, 32(5), 1615-1624.
4. Koshy, T. (2019). Fibonacci and Lucas Numbers with Applications, Volume 2. John Wiley & Sons.
5. Karjanto, N. (2020). Properties of Chebyshev polynomials. arXiv preprint arXiv:2002.01342.
6. Li, Y. (2014). On Chebyshev polynomials, Fibonacci polynomials, and their derivatives. Journal of Applied Mathematics, 2014, Article ID 451953.
7. Li, Y. (2015). Some properties of Fibonacci and Chebyshev polynomials. Advances in Difference Equations, 2015, Article ID 118
8. Mason, J.C., & Handscomb, D.C. (2002). Chebyshev Polynomials (1st ed.). Chapman and Hall/CRC.
9. Włoch, A. (2013). Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers. Applied Mathematics and Computation, 219(10), 5564–5568.
10. Wang, T., & Zhang, H. (2015). Some identities involving the derivative of the first kind Chebyshev polynomials. Mathematical Problems in Engineering, 2015, Article ID 146313.
11. Wu, Z., & Zhang, W. (2012). The sums of the reciprocals of Fibonacci polynomials and Lucas polynomials. Journal of Inequalities and Applications, 2012, Article ID 134.

Manuscript history

• Received: 14 September 2022
• Revised: 22 March 2023
• Accepted: 13 April 2023
• Online First: 16 April 2023

Copyright information

Ⓒ 2023 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).