Cassini and Catalan identities
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:History
Cassini's formula was discovered in 1680 by Giovanni Domenico Cassini, then director of the Paris Observatory, and independently proven by Robert Simson. Eugène Charles Catalan found the identity named after him in 1879.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 th power of a matrix with determinant −1:
