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,

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

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

Catalan's identity generalizes this:

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

Vajda's identity generalizes this:

${\displaystyle F_{n+i}F_{n+j}-F_{n}F_{n+i+j}=(-1)^{n}F_{i}F_{j}.}$

Cassini's formula was discovered in 1680 by Giovanni Domenico Cassini, then director of the Paris Observatory, and independently proven by Robert Simson (1753).^[1] However Johannes Kepler presumably knew the identity already in 1608.^[2]

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 (Catalan 1886). This explains why some give 1879 and others 1886 as the date for Catalan's identity (Tuenter 2022, p. 314).

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.^[3][4] However, the identity had been published earlier in 1960 by Dustan Everman as problem 1396 in The American Mathematical Monthly,^[1] and in 1901 by Alberto Tagiuri in Periodico di Matematica.^[5]

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:

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

Consider the induction statement:

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

The base case ${\displaystyle n=1}$ is true.

Assume the statement is true for ${\displaystyle n}$. Then:

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

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

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

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

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

so the statement is true for all integers ${\displaystyle n>0}$.

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

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

So,

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

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

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

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

Using ${\displaystyle \phi \psi =-1}$,

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

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

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

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

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

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

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

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

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

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

Cancelling the ${\displaystyle 5}$'s gives the result.

• Catalan, Eugène-Charles (December 1886). "CLXXXIX. — Sur la série de Lamé". Mémoires de la Société Royale des Sciences de Liège, Deuxième Série. 13: 319–321.
• 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.
• Tuenter, Hans J. H. (November 2022). "Fibonacci Summation Identities arising from Catalan's Identity". The Fibonacci Quarterly. 60 (4): 312–319. doi:10.1080/00150517.2022.12427460. MR 4539699. Zbl 1512.11025.
• 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.

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