# Cassini and Catalan identities

Cassini's identity and Catalan's identity are mathematical identities for the Fibonacci numbers. Cassini's identity, a special case of Catalan's identity, states that for the nth Fibonacci number,

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

Catalan's identity generalizes this:

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

Vajda's identity generalizes this:

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

## History

Cassini's formula was discovered in 1680 by Jean-Dominique Cassini, then director of the Paris Observatory, and independently proven by Robert Simson (1753). Eugène Charles Catalan found the identity named after him in 1879.

## Proof by matrix theory

A quick proof of Cassini's identity may be given (Knuth 1997, p. 81) by recognising the left side of the equation as a determinant of a 2×2 matrix of Fibonacci numbers. The result is almost immediate when the matrix is seen to be the nth power of a matrix with determinant −1:

${\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}=\det \left[{\begin{matrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{matrix}}\right]=\det \left[{\begin{matrix}1&1\\1&0\end{matrix}}\right]^{n}=\left(\det \left[{\begin{matrix}1&1\\1&0\end{matrix}}\right]\right)^{n}=(-1)^{n}.}$

## References

• Knuth, Donald Ervin (1997), The Art of Computer Programming, Volume 1: Fundamental Algorithms, The Art of Computer Programming, 1 (3rd ed.), Reading, Mass: Addison-Wesley, ISBN 0-201-89683-4