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

Proof

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Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $30$: Solution: $\text{(b)}$