Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers/Lemma 2

Theorem

 * $\ds \sum_{k \mathop \in \Z} \dbinom m k_\FF \paren {-1}^{\ceiling {\paren {m - k} / 2} } {F_{n + k} }^{m - 2} \paren {-1}^k F_{m - k} = \paren {-1}^m F_m \sum_{k \mathop \in \Z} \dbinom {m - 1} k_\FF \paren {-1}^{\ceiling {\paren {m - 1 - k} / 2} } {F_{n + k} }^{m - 2} = 0$

where:
 * $\dbinom m k_\FF$ denotes a Fibonomial coefficient
 * $F_{n + k}$ denotes the $n + k$th Fibonacci number
 * $\ceiling {\, \cdot \,}$ denotes the ceiling function