D'Ocagne's Identity

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Theorem

Let $F_k$ be the $k$th Fibonacci number.


Then:

$\forall m, n \in \Z: \paren {-1}^n F_{m - n} = F_m F_{n + 1} - F_n F_{n - 1}$


Proof

\(\ds \paren {-1}^n F_i F_j\) \(=\) \(\ds F_{n + i} F_{n + j} - F_n F_{n + i + j}\) Vajda's Identity
\(\ds \leadsto \ \ \) \(\ds \paren {-1}^n F_{m - n} F_1\) \(=\) \(\ds F_{n + \paren {m - n} } F_{n + 1} - F_n F_{n + \paren {m - n} + 1}\) setting $i \gets m - n$ and $j \gets 1$
\(\ds \leadsto \ \ \) \(\ds \paren {-1}^n F_{m - n}\) \(=\) \(\ds F_m F_{n + 1} - F_n F_{m + 1}\) Definition of Fibonacci Numbers: $F_1 = 1$

$\blacksquare$


Source of Name

This entry was named for Philbert Maurice d'Ocagne.