On the properties of generalized Fibonacci like polynomials

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 {Mn}(x ), with initial conditions M0(x ) = 2 and M1(x ) = m (x ) + k (x ), which is generated by the recurrence relation Mn+1(x ) = k (x )Mn (x )+Mn−1(x ) for n ≥ 2, where k (x ), m (x ) are polynomials with real coefficients.We produce an extended Binet’s formula for {Mn}(x ) and,thereby identities such as Simpson’s,Catalan’s,d’Ocagene’s,etc.using matrix algebra.Moreover, we present sum formulas concerning this new generalization. MSC: 11B39 • 11B83