Cassini Identity

__notoc__

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 ''n''th Fibonacci number,
:$F_F_ - F_n^2 = (-1)^n.$

Note here $F_0$ is taken to be 0, and $F_1$ is taken to be 1.

Catalan's identity generalizes this:
:$F_n^2 - F_F_ = (-1)^F_r^2.$

Vajda's identity generalizes this:
:$F_F_ - F_F_ = (-1)^nF_F_.$

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.Douglas B. West: ''Combinatorial Mathematics''. Cambridge University Press, 2020, p
61
/ref>Steven Vadja: ''Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications''. Dover, 2008, , p. 28 (original publication 1989 at Ellis Horwood) However the identity was already published in 1960 by Dustan Everman as problem 1396 in The American Mathematical Monthly.Thomas Koshy: ''Fibonacci and Lucas Numbers with Applications''. Wiley, 2001, , pp. 74-75, 83, 88

Proof of Cassini identity

Proof by matrix theory

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 th power of a matrix with determinant −1:
:F_F_ - F_n^2
=\det\left beginF_&F_n\\F_n&F_\end\right=\det\left begin1&1\\1&0\end\rightn
=\left(\det\left begin1&1\\1&0\end\rightright)^n
=(-1)^n.

Proof by induction

Consider the induction statement:

:$F_F_ - F_n^2 = (-1)^n$

The base case $n=1$ is true.

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

:$F_F_ - F_n^2 + F_nF_ - F_nF_ = (-1)^n$

:$F_F_ + F_nF_ - F_n^2 - F_nF_ = (-1)^n$

:$F_(F_ + F_n) - F_n(F_n + F_) = (-1)^n$

:$F_^2 - F_nF_ = (-1)^n$

:$F_nF_ - F_^2 = (-1)^$

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

Proof of Catalan identity

We use Binet's formula, that $F_n=\frac$, where $\phi=\frac$ and $\psi=\frac$.

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

So,

:$5(F_n^2 - F_F_)$

:$= (\phi^n-\psi^n)^2 - (\phi^-\psi^)(\phi^-\psi^)$

:$= (\phi^ - 2\phi^\psi^ +\psi^) - (\phi^ - \phi^\psi^(\phi^\psi^+\phi^\psi^) + \psi^)$

:$= - 2\phi^\psi^ + \phi^\psi^(\phi^\psi^+\phi^\psi^)$

Using $\phi\psi=-1$,

:$= -(-1)^n2 + (-1)^n(\phi^\psi^+\phi^\psi^)$

and again as $\phi=\frac$,

:$= -(-1)^n2 + (-1)^(\psi^+\phi^)$

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

:$= -(-1)^n2 + (-1)^L_$

Because $L_ = 5 F_n^2 + 2(-1)^n$

:$= -(-1)^n2 + (-1)^(5 F_r^2 + 2(-1)^r)$

:$= -(-1)^n2 + (-1)^2(-1)^r + (-1)^5 F_r^2$

:$= -(-1)^n2 + (-1)^n2 + (-1)^5 F_r^2$

:$= (-1)^5 F_r^2$

Cancelling the $5$'s gives the result.

Notes

References

*

*

*{{cite journal
, last1 = Werman
, first1 = M.
, authorlink2 = Doron Zeilberger
, last2 = Zeilberger
, first2 = D.
, title = A bijective proof of Cassini's Fibonacci identity
, journal = Discrete Mathematics
, volume = 58
, issue = 1
, year = 1986
, pages = 109
, mr = 0820846
, doi = 10.1016/0012-365X(86)90194-9, doi-access = free

External links

Proof of Cassini's identityProof of Catalan's Identity

Mathematical identities
Fibonacci numbers
Articles containing proofs
Giovanni Domenico Cassini