Cassini and Catalan identities

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 "n"th Fibonacci number,

:$F\_\{n-1\}F\_\{n+1\}\; -\; F\_n^2\; =\; (-1)^n.,$

Catalan's identity generalizes this:

:$F\_n^2\; -\; F\_\{n-r\}F\_\{n+r\}\; =\; (-1)^\{n-r\}F\_r^2.,$

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 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 "n"th power of a matrix with determinant −1::

References

*cite journal
author = Simson, R.
authorlink = Robert Simson
title = An Explication of an Obscure Passage in Albert Girard’s Commentary upon Simon Stevin’s Works
journal = Philosophical Transactions of the Royal Society of London
volume = 48
year = 1753
pages = 368–376
doi = 10.1098/rstl.1753.0056

*cite journal
author = Werman, M.; Zeilberger, D.
title = A bijective proof of Cassini's Fibonacci identity
journal = Discrete Mathematics
volume = 58
issue = 1
year = 1986
pages = 109
id = MathSciNet | id = 0820846
doi = 10.1016/0012-365X(86)90194-9

External links

*planetmath reference|id=6382|title=Proof of Cassini's identity
*planetmath reference|id=6389|title=Proof of Catalan's Identity
* [http://milan.milanovic.org/math/english/pi/cassini.htm Cassini formula for Fibonacci numbers]
* [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibFormulae.html Fibonacci and Phi Formulae]