Cassini and Catalan identities

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Mathematical identities for the Fibonacci numbers
Cassini's identity (sometimes called Simson'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,

Note here is taken to be 0, and is taken to be 1. Catalan's identity generalizes this:

Vajda's identity generalizes this:

Cassini's formula was discovered in 1680 by Giovanni Domenico Cassini, then director of the Paris Observatory, and independently proven by Robert Simson (1753). Catalan's identity is named after Eugène Catalan (1814–1894). It can be found in one of his private research notes, entitled "Sur la série de Lamé" and dated October 1879. However, the identity did not appear in print until December 1886 as part of his collected works This explains why some give 1879 and others 1886 as the date for Catalan's identity
The Hungarian British mathematician Steven Vajda (1901–95) published a book on Fibonacci numbers (Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, 1989) which contains the identity carrying his name. However, the identity had been published earlier in 1960 by Dustan Everman as problem 1396 in The American Mathematical Monthly, and in 1901 by Alberto Tagiuri in Periodico di Matematica.

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 nth power of a matrix with determinant −1: