Solving some generalized Vandermonde systems and inverse of their associate matrices via new approaches for the Binet formula

The aim of this paper is to exhibit new expressions for the Binet formula, without solving any type of Vandermonde systems. This allows us to establish a tractable formula for generalized Fibonacci sequences. As a consequence, new solutions of the Vandermonde systems and some special generalized Vandermonde systems are provided, moreover the inverses of their associated matrices are explicated. Some new results are formulated, also others are recovered and generalized. Finally, comparisons with other methods are broached and some numerical examples are supplied.