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

Catalan's identity generalizes this:

Vajda's identity generalizes this:

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HistoryEdit

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 theoryEdit

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:

 

ReferencesEdit

  • 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

External linksEdit