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)^{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 $n$ th 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

Knuth, Donald Ervin (1997), The Art of Computer Programming, Volume 1: Fundamental Algorithms, The Art of Computer Programming, 1 (3rd ed.), Reading, Mass: Addison-Wesley, ISBN 0-201-89683-4 .

Simson, R. (1753). "An Explication of an Obscure Passage in Albert Girard's Commentary upon Simon Stevin's Works". Philosophical Transactions of the Royal Society of London. 48 (0): 368–376. doi:10.1098/rstl.1753.0056. .

Werman, M.; Zeilberger, D. (1986). "A bijective proof of Cassini's Fibonacci identity". Discrete Mathematics. 58 (1): 109. doi:10.1016/0012-365X(86)90194-9. MR 0820846.

External links

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