Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers

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Theorem

Let $m \in \Z_{>0}$ be a (strictly) positive integer.


Then:

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

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


Proof

Lemma 1

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


Lemma 2

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



Examples

Example: $m = 3$

${F_n}^n - 2 {F_{n + 1} }^2 - 2 {F_{n + 2} }^2 + {F_{n + 3} }^2 = 0$


Historical Note

Donald E. Knuth, in his The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed. of $1997$, attributes this result to Dov Jarden and Theodore Samuel Motzkin.

However, he provides a citation for Jarden's work only, and it is unclear from where Motzkin's contribution is sourced.


Sources