Sums of fourth powers of Fibonacci and Lucas numbers

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.

9 pages