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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,

Contents

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 .

Vajda's identity generalizes this:

F n + i F n + j − F n F n + i + j = ( − 1 ) n F i F j .

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 (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:

F n − 1 F n + 1 − F n 2 = det [ F n + 1 F n F n F n − 1 ] = det [ 1 1 1 0 ] n = ( det [ 1 1 1 0 ] ) n = ( − 1 ) n .

References

Cassini and Catalan identities Wikipedia
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