Series and sums involving the floor function

Abstract: Let $(a_n){n\geq 0}$ be an arbitrary sequence and $(a{\lfloor n/k \rfloor}){n\geq 0}$ its dual floor sequence. We study infinite series and finite generalized binomial sums involving $(a{\lfloor n/k \rfloor})_{n\geq 0}$. As applications we prove a range of new closed form expressions for Fibonacci (Lucas) series and binomial sum identities as particular cases.