Cassini's Identity/Negative Indices

Theorem

Let $n \in \Z_{<0}$ be a negative integer.

Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers).

Then Cassini's Identity:

$F_{n + 1} F_{n - 1} - F_n^2 = \paren {-1}^n$

continues to hold.

Proof

Let $n \in \Z_{> 0}$.

Then:

\(\ds F_{-\paren {n + 1} } F_{-\paren {n - 1} } - {F_{-n} }^2\)

\(=\)

\(\ds \paren {-1}^{n + 2} F_{n + 1} \paren {-1}^n F_{n - 1} - \paren {\paren {-1}^{n + 1} F_n}^2\)

Fibonacci Number with Negative Index

\(\ds \)

\(=\)

\(\ds \paren {-1}^{2 n + 2} F_{n + 1} F_{n - 1} - \paren {-1}^{2 n + 2} F_n^2\)

\(\ds \)

\(=\)

\(\ds F_{n + 1} F_{n - 1} - F_n^2\)

$-1$ to an even power is $1$

$\blacksquare$

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 $9$