Cassini and Catalan identities
Encyclopedia
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,
Catalan's identity generalizes this:
, then director of the Paris Observatory, and independently proven by Robert Simson
(1753). Eugène Charles Catalan
found the identity named after him in 1879.
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:
History
Cassini's formula was discovered in 1680 by Jean-Dominique CassiniGiovanni Domenico Cassini
This article is about the Italian-born astronomer. For his French-born great-grandson, see Jean-Dominique Cassini.Giovanni Domenico Cassini was an Italian/French mathematician, astronomer, engineer, and astrologer...
, then director of the Paris Observatory, and independently proven by Robert Simson
Robert Simson
Robert Simson was a Scottish mathematician and professor of mathematics at the University of Glasgow. The pedal line of a triangle is sometimes called the "Simson line" after him.-Life:...
(1753). Eugène Charles Catalan
Eugène Charles Catalan
Eugène Charles Catalan was a French and Belgian mathematician.- Biography :Catalan was born in Bruges , the only child of a French jeweller by the name of Joseph Catalan, in 1814. In 1825, he traveled to Paris and learned mathematics at École Polytechnique, where he met Joseph Liouville...
found the identity named after him in 1879.
Proof by matrix theory
A quick proof of Cassini's identity may be given by recognising the left side of the equationSides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...
as a determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
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: