Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Lemma 1

Theorem

 * $\displaystyle \sum_{k \mathop \in \Z} \dbinom m k_\mathcal F \left({-1}\right)^{\left\lceil{\left({m - k}\right) / 2}\right\rceil} {F_{n + k} }^{m - 2} F_k = F_m \sum_{k \mathop \in \Z} \dbinom {m - 1} {k - 1}_\mathcal F \left({-1}\right)^{\left\lceil{\left({m - k}\right) / 2}\right\rceil} {F_{n + k} }^{m - 2} = 0$

where:
 * $\dbinom m k_\mathcal F$ denotes a Fibonomial coefficient
 * $F_{n + k}$ denotes the $n + k$th Fibonacci number
 * $\left\lceil{\, \cdot \,}\right\rceil$ denotes the ceiling function