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,
Catalan's identity generalizes this:
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[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.](../../../../math/5/a/2/5a2576641952b0434c3dd420616483d5.png)
[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.
- Werman, N.; Zeilberger, D. (1986). "A bijective proof of Cassini's Fibonacci identity". Discrete Mathematics 58 (1): 109. doi:. MR0820846.
