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:

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

Vajda's identity generalizes this:

$F_{n+i}F_{n+j} - F_{n}F_{n+i+j} = (-1)^nF_{i}F_{j}.\,$

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:

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

References

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External links

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