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 nth Fibonacci number,

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

Catalan's identity generalizes this:

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

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 nth power of a matrix with determinant −1:

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

References

• Simson, R.; Philip, H. (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 (0): 368–376. doi:10.1098/rstl.1753.0056. 

• Werman, M.; Zeilberger, D. (1986). "A bijective proof of Cassini's Fibonacci identity". Discrete Mathematics 58 (1): 109. doi:10.1016/0012-365X(86)90194-9. MR0820846. 

External links

• Proof of Cassini's identity on PlanetMath
• Proof of Catalan's Identity on PlanetMath
• Cassini formula for Fibonacci numbers
• Fibonacci and Phi Formulae