Cassini and Catalan identities
Catalan's identity generalizes this:
Vajda's identity generalizes this:
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:
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (June 2011)|
- 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.
- 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. MR 0820846.