New Generalizations of Fibonacci and Lucas Polynomials

Abstract

We consider the polynomials f_n(x)  and l_n(x)   which are generated by the recurrence relations f_n(x) = 2ax f_n-1(x)+(b-a²) f_n-2(x)   for n >=2  and l_n(x) = 2ax l_n-1(x)+(b-a²) l_n-2(x)  for n >=2 with the initial conditions f₀(x) =0 , f₁(x) =1 and  l₀(x) =2 , l₁(x) =2ax  where  a and  b are any non-zero real numbers. We obtain the new generalizations of Fibonacci and Lucas polynomials. Moreover, we obtain generating functions, Binet’s formulas and some identities involving  f_n(x)  and l_n(x) .