Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) 234 - 251 (ISSN: 2347-2529) Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics On the properties of generalized Fibonacci like polynomials Research Article A. D. Godase 1, ∗ , M. B. Dhakne 2 1 Department of Mathematics, V. P. College,Vaijapur, India 2 Department of Mathematics, Dr. B. A. M. University,Aurangabad, India Received 15 November 2014; accepted (in revised version) 17 March 2015 Abstract: The Fibonacci polynomial has been generalized in many ways,some by preserving the initial conditions,and others by preserving the recurrence relation.In this article,we study new generalization {M n }( x ), with initial conditions M 0 ( x )= 2 and M 1 ( x )= m( x )+ k ( x ), which is generated by the recurrence relation M n+1 ( x )= k ( x )M n ( x )+ M n−1 ( x ) for n ≥ 2, where k ( x ), m( x ) are polynomials with real coefficients.We produce an extended Binet’s formula for {M n }( x ) and,thereby identities such as Simpson’s,Catalan’s,d’Ocagene’s,etc.using matrix alge- bra.Moreover, we present sum formulas concerning this new generalization. MSC: 11B39 • 11B83 Keywords: Fibonacci polynomials • Lucas polynomials • Recurrence relation • Matrix algebra c  2015 IJAAMM all rights reserved. 1. Introduction Definition 1.1. The Fibonacci polynomial F n ( x ), n = 1, 2, 3, ... is defined as, F n +1 ( x )= k ( x )F n ( x )+ F n −1 ( x ) with F 0 ( x )= 0, F 1 ( x )= 1 for n ≥ 1. Definition 1.2. The Lucas polynomial L n ( x ), n = 1, 2, 3, ... is defined as, L n +1 ( x )= k ( x )L n ( x )+ L n −1 ( x ) with L 0 ( x )= 2, L 1 ( x )= k ( x ) for n ≥ 1. In this paper, we study the new generalization {M n ( x )}, with initial conditions M 0 ( x )= 2 and M 1 ( x )= m ( x )+ k ( x ), which is generated by the recurrence relation, M n +1 ( x )= k ( x )M n ( x )+ M n −1 ( x ) for , n ≥ 2, where k ( x ), m ( x ) are polynomials with real coefficients. Many authors have studied relationship between the Fibonacci sequence,its certain generalizations and matrix properties (for more details see [1]-[11] and Fibonacci polynomials.In this paper we define a new generalization of the Fibonacci polynomial and give identities and sum formulas concerning this new generalization. ∗ Corresponding author. E-mail address: ashokgodse2012@gmail.com 234