A Study on Generalized Fibonacci Numbers: Sum Formulas $\sum_{k=0}^{n}kx^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}kx^{k}W_{-k}^{3}$ for the Cubes of Terms

Abstract

In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}kx^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}kx^{k}W_{-k}^{3}$ for the cubes of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers.