Cassini and Catalan identities

Cassini's identity (sometimes called Simson's identity), Catalan's identity and Vajda's identity are mathematical identities for the Fibonacci numbers. Cassini's identity, a special case of the other two, states that for the nth Fibonacci number,

${\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}.}$

Note here ${\displaystyle F_{0}}$ is taken to be 0, and ${\displaystyle F_{1}}$ is taken to be 1.

Catalan's identity generalizes this to allow ${\displaystyle r\neq 1}$:

${\displaystyle F_{n}^{2}-F_{n-r}F_{n+r}=(-1)^{n-r}F_{r}^{2}.}$

Vajda's identity further generalizes this to allow ${\displaystyle m\neq n}$:

${\displaystyle F_{m}F_{n}-F_{m-r}F_{n+r}=\left({-1}\right)^{m-r}F_{r}F_{r+n-m},}$

also written as:

${\displaystyle F_{n+i}F_{n+j}-F_{n}F_{n+i+j}=(-1)^{n}F_{i}F_{j}.}$