Sums of fourth powers of Fibonacci and Lucas numbers

TL;DR

This paper derives closed-form formulas for sums of fourth powers of Fibonacci and Lucas numbers, including their alternating sums, expanding the mathematical understanding of these sequences.

Contribution

It provides new explicit formulas for sums of fourth powers of Fibonacci and Lucas numbers, including alternating sums, which were not previously available.

Findings

Closed-form expressions for th powers sums of Fibonacci numbers

Closed-form expressions for th powers sums of Lucas numbers

Extension of previous work on alternating sums

Abstract

We obtain closed-form expressions for all sums of the form \mbox{$\sum_{k = 1}^n {F_{mk}{}^4 }$} and \mbox{$\sum_{k = 1}^n {L_{mk}{}^4 }$} and their alternating versions, where $F_i$ and $L_i$ denote Fibonacci and Lucas numbers respectively. Our results complement those of Melham who studied the alternating sums.