Cassini and Catalan identities

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,

$F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}.$

Note here $F_{0}$ is taken to be 0, and $F_{1}$ is taken to be 1.

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 Giovanni Domenico Cassini, then director of the Paris Observatory, and independently proven by Robert Simson (1753). However Johannes Kepler presumably knew the identity already in 1608. Eugène Charles Catalan found the identity named after him in 1879. The 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 was already published in 1960 by Dustan Everman as problem 1396 in The American Mathematical Monthly.

Proof of Cassini identity

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:

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

Proof by induction

Consider the induction statement:

$F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}$

The base case $n=1$ is true.

Assume the statement is true for $n$. Then:

$F_{n-1}F_{n+1}-F_{n}^{2}+F_{n}F_{n+1}-F_{n}F_{n+1}=(-1)^{n}$

$F_{n-1}F_{n+1}+F_{n}F_{n+1}-F_{n}^{2}-F_{n}F_{n+1}=(-1)^{n}$

$F_{n+1}(F_{n-1}+F_{n})-F_{n}(F_{n}+F_{n+1})=(-1)^{n}$

$F_{n+1}^{2}-F_{n}F_{n+2}=(-1)^{n}$

$F_{n}F_{n+2}-F_{n+1}^{2}=(-1)^{n+1}$

so the statement is true for all integers $n>0$.

Proof of Catalan identity

We use Binet's formula, that $F_{n}={\frac {\phi ^{n}-\psi ^{n}}{\sqrt {5}}}$, where $\phi ={\frac {1+{\sqrt {5}}}{2}}$ and $\psi ={\frac {1-{\sqrt {5}}}{2}}$.

Hence, $\phi +\psi =1$ and $\phi \psi =-1$.

So,

$5(F_{n}^{2}-F_{n-r}F_{n+r})$

$=(\phi ^{n}-\psi ^{n})^{2}-(\phi ^{n-r}-\psi ^{n-r})(\phi ^{n+r}-\psi ^{n+r})$

$=(\phi ^{2n}-2\phi ^{n}\psi ^{n}+\psi ^{2n})-(\phi ^{2n}-\phi ^{n}\psi ^{n}(\phi ^{-r}\psi ^{r}+\phi ^{r}\psi ^{-r})+\psi ^{2n})$

$=-2\phi ^{n}\psi ^{n}+\phi ^{n}\psi ^{n}(\phi ^{-r}\psi ^{r}+\phi ^{r}\psi ^{-r})$

Using $\phi \psi =-1$,

$=-(-1)^{n}2+(-1)^{n}(\phi ^{-r}\psi ^{r}+\phi ^{r}\psi ^{-r})$

and again as $\phi ={\frac {-1}{\psi }}$,

$=-(-1)^{n}2+(-1)^{n-r}(\psi ^{2r}+\phi ^{2r})$

The Lucas number $L_{n}$ is defined as $L_{n}=\phi ^{n}+\psi ^{n}$, so

$=-(-1)^{n}2+(-1)^{n-r}L_{2r}$

Because $L_{2n}=5F_{n}^{2}+2(-1)^{n}$

$=-(-1)^{n}2+(-1)^{n-r}(5F_{r}^{2}+2(-1)^{r})$

$=-(-1)^{n}2+(-1)^{n-r}2(-1)^{r}+(-1)^{n-r}5F_{r}^{2}$

$=-(-1)^{n}2+(-1)^{n}2+(-1)^{n-r}5F_{r}^{2}$

$=(-1)^{n-r}5F_{r}^{2}$

Cancelling the $5$'s gives the result.

References

• Knuth, Donald Ervin (1997), The Art of Computer Programming, Volume 1: Fundamental Algorithms, The Art of Computer Programming, vol. 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: 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

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