On some properties of generalized Fibonacci polynomials

Fibonacci polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions and by varying the recurrence relation and maintaining the initial conditions. In this paper, both the recurrence relation and initial conditions of generalized Fibonacci polynomials are varied and defined by recurrence relation as R n ( x ) = a x R n − 1 ( x ) + b R n − 2 ( x ) for all n ≥ 2 , with initial conditions R 0 ( x ) = 2 p and R 1 ( x ) = p x + q where a and b are positive integers and p and q are non-negative integers. Further some fundamental properties of these generalized polynomials such as explicit sum formula, sum of first n terms, sum of first n terms with (odd or even) indices and generalized identity are derived by Binet’s formula and generating function only.