Cassini and Catalan identities

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

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

Catalan's identity generalizes this:

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

1 History
2 Proof by matrix theory
3 References
4 External links

[edit] 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.

[edit] Proof by matrix theory

A quick proof of Cassini's identity may be given 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:

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

[edit] References

Simson, R. (1753). "An Explication of an Obscure Passage in Albert Girard’s Commentary upon Simon Stevin’s Works". Philosophical Transactions of the Royal Society of London 48: 368–376.