Abstract
Various sequences of polynomials by the names of Fibonacci and Lucas polynomials occur in the literature over a century. Generalization of the Fibonacci polynomial has been done using various approaches. One usually found in the literature that the generalization is done by varying the initial conditions. In this paper we study the so-called generalized Fibonacci polynomials: with and where is any integer. Further we give some fundamental properties about the generalized Fibonacci polynomials.

Keywords
Fibonacci polynomials, Generalized Fibonacci polynomials, Generating function, Binet’s Formula

Cite this paper
Omprakash Sikhwal, Yashwant Vyas, Generalized Fibonacci Polynomials and Some Fundamental Properties , SCIREA Journal of Mathematics. Volume 1, Issue 1, October 2016 | PP. 16-23.

References

[ 1 ]	A. R. Glasson, “Remainder Formulas, Involving Generalized Fibonacci and Lucas Polynomials,” The Fibonacci Quarterly, 33 (3), 268-172, 1965.
[ 2 ]	A. F. Horadam and J. M. Mahon, “Pell and Pell-Lucas Polynomials,” The Fibonacci Quarterly, 23 (10), 7-20, 1985.
[ 3 ]	A. Lupas, “A Guide of Fibonacci And Lucas Polynomials,” Octagon Mathematics Magazine, 7 (1), 2-12, 1999.
[ 4 ]	A. Nalli and P. Haukkanen, “On generalized Fibonacci and Lucas polynomials,” Chaos, Solitons and Fractals, 42(5), 3179–3186, 2009.
[ 5 ]	B. G. S. Doman and J. K. Williams, “Fibonacci and Lucas Polynomials,” Mathematical Proceedings of the Cambridge Philosophical Society, 90, Part 3, 385-387, 1981.
[ 6 ]	Beverage David, “A Polynomial Representation of Fibonacci Numbers,” The Fibonacci Quarterly, 9, 541-544, 1971.
[ 7 ]	B. Singh, O. Sikhwal and Y. K. Panwar, “Generalized Determinantal Identities Involving Lucas Polynomials,” Applied Mathematical Sciences, 3 (8), 377-388, 2009.
[ 8 ]	B. Singh, S. Bhatnagar and O. Sikhwal, “Fibonacci-Like Polynomials and Some identities,” International Journal of Advanced Mathematical Sciences, 1 (3), 152-157, 2013.
[ 9 ]	E. C. Catalan, “Notes Surla Theorie des Fractions continuess et Sur Certaines Series,” Mem. Acad. R. Belgique, 45, 1-82, 1883.
[ 10 ]	E. Jacosthal, “Fibonacci Polynome und Kreisteil Ungsgleichugen Sitzungsberichte der Berliner,” Math. Gesellschaft, 17, 43-57, 1919-20.
[ 11 ]	G. B. Djordjevi and H. M. Srivastava, “Some generalizations of the incomplete Fibonacci and the incomplete Lucas polynomials,” Adv. Stud. Contemp. Math., 11, 11-32, 2005.
[ 12 ]	J. Ivie, “Ageneral 𝑄-matrix,” The Fibonacci Quarterly, 10 (3), 255–261, 1972.
[ 13 ]	Marjorie Bicknell, “A Primer for the Fibonacci Numbers: Part VII- An Introduction to Fibonacci Polynomials and their Divisibility Properties,” The Fibonacci Quarterly, 8 (4), 407-420, 1970.
[ 14 ]	M. Singh, M., O. Sikhwal and Y. Gupta., “Generalized Fibonacci-Lucas Polynomials,” International Journal of Advanced Mathematical Sciences, 2 (1), 81-87, 2014.
[ 15 ]	M. N. S. Swamy, “Generalized Fibonacci and Lucas Polynomials and their associated diagonal Polynomials,” The Fibonacci Quarterly, 37, 213-222, 1999..
[ 16 ]	O. Sikhwal, “Generalization of Fibonacci Polynomials,” Kartindo Academia, Kartindo.com: International Publisher and distributor, India 2014.
[ 17 ]	P. F. Byrd, “Expansion of Analytic Functions in Polynomials Associated with Fibonacci Numbers,” The Fibonacci Quarterly, 1 (1), 16-29, 1963.
[ 18 ]	S. L. Basin, ‘The appearance of Fibonacci Numbers and the Q matrix in Electrical Network Theory,” Mathematics Magazine, 36 (2), 84-97, 1963.
[ 19 ]	T. Koshy, Fibonacci and Lucas Numbers With Applications, John Wiley and Sons, New York, 2001.
[ 20 ]	V. E. Hoggatt Jr. and M. Bicknell, “Generalized Fibonacci polynomials and Zeckendorf ’s theorem,” The Fibonacci Quarterly, 11 (4), 399–419, 1973.
[ 21 ]	V. E. Hoggatt Jr. and C. T. Long, “Divisibility Properties of Fibonacci Polynomials,” The Fibonacci Quarterly,12 (2), 113-120. 1974.
[ 22 ]	W. A. Webb and E. A. Parberry, “Divisibility Properties of Fibonacci Polynomials,” The Fibonacci Quarterly, 7(5), 457-463, 1969.