Factored closed-form expressions for the sums of cubes of Fibonacci and Lucas numbers

TL;DR

This paper derives explicit factored closed-form formulas for sums of cubes of Fibonacci and Lucas numbers, expanding mathematical understanding of these sequences and their summations.

Contribution

It introduces new explicit factored closed-form expressions for sums of cubes of Fibonacci and Lucas numbers for arbitrary integer parameters.

Findings

Derived closed-form formulas for Fibonacci sum of cubes.

Derived closed-form formulas for Lucas sum of cubes.

Formulas are explicitly factored and applicable for all integer parameters.

Abstract

We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of the form \mbox{$\sum_{k = 1}^n {F_{2rk}^3 }$} and \mbox{$\sum_{k = 1}^n {L_{2rk}^3 }$}, where $r$~and~$n$ are integers.