Integration Sequences of Fibonacci and Lucas Polynomials

Abstract

Let us consider the Fibonacci polynomials U n (x)= W n (0,1; x, - 1) and the Lucas polynomials V n (x) = W n (2,x; x, - 1) (See [6] and [7] for background material on the sequences W n ) defined as $$ {U_n}(x) = x{U_{{n - 1}}}(x) + {U_{{n - 2}}}(x)\;\;\left[ {{U_0}(x) = 0,\;{U_1}(x) = 1} \right] $$ (1.1) and $$ {V_n}(x) = x{V_{{n - 1}}}(x) + {V_{{n - 2}}}(x)\;\;\left[ {{V_0}(x) = 2,\;{V_1}(x) = x} \right], $$ (1.2) , where x is an indeterminate. It is well-known that the quantities U n (x) and V n (x) can also be expressed by means of the Binet forms $$ {U_n}(x) = \left( {\alpha_x^n - \beta_x^n} \right)/{\Delta_x} $$ (1.3) and $$ {V_n}(x) = \alpha_x^n + \beta_x^n, $$ (1.4) where $$ {\Delta_x} = \sqrt {{{x^2} + 4}}, $$ (1.5) $$ {\alpha_x} = \left( {x + {\Delta_x}} \right)/2, $$ (1.6) $$ {\beta_x} = - 1/{\alpha_x} = x - {\alpha_x} = \left( {x - {\Delta_x}} \right)/2. $$ (1.7) .