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Complex factorizations of the generalized Fibonacci sequences $\{q_n\}$
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In this note, we consider a generalized Fibonacci sequence $\{q_n\}$. Then give a connection between the sequence $\{q_n\}$ and the Chebyshev polynomials of the second kind $U_n(x)$. With the aid of factorization of Chebyshev polynomials of the second kind $U_n(x)$, we derive the complex factorizations of the sequence $\{q_n\}$.
Abstract
In this note, we consider a generalized Fibonacci sequence $\{q_n\}$. Then give a connection between the sequence $\{q_n\}$ and the Chebyshev polynomials of the second kind $U_n(x)$. With the aid of factorization of Chebyshev polynomials of the second kind $U_n(x)$, we derive the complex factorizations of the sequence $\{q_n\}$.
참고문헌 (12)
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- Y. K. Gupta, "Generalized Fibonacci-Like Sequence Associated with Fibonacci and Lucas Sequences", Turkish Journal of Analysis and Number Theory, 2(6), pp.233~238 (2014)
- A. F. Horadam, "A generalized Fibonacci sequences", Amer. Math. Monthly, 68, pp.455~459 (1961)
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- 장영호, "Linearlization of generalized Fibonacci sequences", 한국수학논문집, 22(3), pp.443~454 (2014), 강원경기수학회[DOI]
- D. Kalman, "The Fibonacci numbers - Exposed", The Mathematical Magazine, 2 (2002)
- A. Oteles, "Positive integer power of certain complex tridiago-nal matrices", Applied Mathematics and Computation, 219(21), pp.10448~10455 (2013)
- T.J. Rivlin, "The Chebyshev Polynomials-From Approximation Theory to Algebra and Number Theory" (1990), Wiley-Interscience, John Wiley
- O. Yayenie, "A note on generalized Fibonacci sequences", Applied Mathematics and Computation, 217, pp.5603~5611 (2011)
- H. Zhang, "On the reciprocal sums of the generalized Fibonacci se-quences", Adv. Di er. Equ, 377 (2013)
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