On some properties of generalized Fibonacci and Lucas polynomials

In this paper we investigate some properties of generalized Fibonacci and Lucaspolynomials. We give some new identities using matrices and Laplace expansionfor the generalized Fibonacci and Lucas polynomials. Also, we introducenew families of tridiagonal matrices whose successive determinants generate any subsequence of these polynomials.

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[1] Cahill, N. D., D'Errico, J. R., Spence, J. S., Complex factorizations of the Fibonacci and Lucas numbers. Fibonacci Quart., 41, 1, 13-19, (2003).
[2] Catarino, P., On some idenitities for k-Fibonacci sequence. Int. J. Contemp. Math. Sciences, 9, 1, 37-42, (2014).
[3] Doman, B. G. S., Generating functions for Chebyshev polynomials. Int. J. Pure Appl. Math. 63, 2, 197-205, (2010).
[4] Falcon, S., Plaza, A., On the Fibonacci k-numbers. Chaos Solitons Fractals. 32, 5, 1615-1624, (2007).
[5] Falcon, S., On the k-Lucas numbers. Int. J. Contemp. Math. Sci. 6, 21-24, 1039-1050, (2011).
[6] Feng, J., Fibonacci identities via the determinant of tridiagonal matrix. Appl. Math. Comput. 217, 12, 5978-5981, (2011).
[7] Godase, A. D., Dhakne, M. B., On the properties of k-Fibonacci and k-Lucas numbers. Int. J. Adv. Appl. Math. Mech. 2, 1, 100-106, (2014).
[8] Kalman, D., Mena, R., The Fibonacci NumbersExposed. Math. Mag. 76, 3, 167-181, (2003).
[9] Kılı¸c, E., Stanica, P., Factorizations and representations of binary polynomial recurrences by matrix methods. Rocky Mountain J. Math. 41, 4, 1247-1264, (2011).
[10] Koshy, T., Fibonacci and Lucas numbers with applications. Pure and Applied Mathematics (New York). Wiley-Interscience, New York, (2001).
[11] Lee, G. Y., Asci, M. Some properties of the (p, q)- Fibonacci and (p, q)- Lucas polynomials. J. Appl. Math., (2012).
[12] Mason, J. C., Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration and integral transforms. Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991). J. Comput. Appl. Math. 49, 1-3, 169-178, (1993).
[13] Mason, J. C., Handscomb, D. C., Chebyshev polynomials. Chapman&Hall/CRC, Boca Raton, FL, 2003. xiv+341 pp.
[14] Meyer, C., Matrix analysis and applied linear algebra, With 1 CD-ROM (Windows, Macintosh and UNIX) and a solutions manual (iv+171 pp.). Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (2000). xii+718 pp.
[15] Ozg¨ur, N. Y., U¸car, S., ¨ Oztun¸c, ¨ O. Complex Factor- ¨ izations of the k-Fibonacci and k-Lucas numbers. An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 62, 1, 13-20, (2016).
[16] Nalli, A., Haukkanen, P., On generalized Fibonacci and Lucas polynomials. Chaos Solitons Fractals 42, 5, 3179-3186, (2009).
[17] Rivlin, J. T., The Chebyshev polynomials. Pure and Applied Mathematics, Wiley-Interscience [John Wiley&Sons], New York-London-Sdyney, (1974). vi+186 pp.
[18] S¸iar, Z., Keskin, R., Some new identites concerning Generalized Fibonacci and Lucas Numbers. Hacet. J. Math. Stat. 42, 3, 211-222, (2013).
[19] Wang, J., Some new results for the (p, q)- Fibonacci and Lucas polynomials. Adv Differ Equ. 64, (2014).