Cassini and Catalan identities

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https://en.wikipedia.org/wiki/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,

Catalan's identity generalizes this:

Vajda's identity generalizes this:

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). [1] However Johannes Kepler presumably knew the identity already in 1608. [2] Eugène Charles Catalan found the identity named after him in 1879. [1] 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. [3] [4] However the identity was already published in 1960 by Dustan Everman as problem 1396 in The American Mathematical Monthly. [1]

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:

Proof by induction

Consider the induction statement:

The base case is true.

Assume the statement is true for . Then:

so the statement is true for all integers .

Proof of Catalan identity

We use Binet's Theorem, that , where and .

Hence, and .

So,

Using ,

and again as ,

The Lucas number is defined as , so

Because

Cancelling the 's gives the result.

Notes

  1. ^ a b c Thomas Koshy: Fibonacci and Lucas Numbers with Applications. Wiley, 2001, ISBN  9781118031315, pp. 74-75, 83, 88
  2. ^ Miodrag Petkovic: Famous Puzzles of Great Mathematicians. AMS, 2009, ISBN  9780821848142, S. 30-31
  3. ^ Douglas B. West: Combinatorial Mathematics. Cambridge University Press, 2020, p. 61
  4. ^ Steven Vadja: Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications. Dover, 2008, ISBN  978-0486462769, p. 28 (original publication 1989 at Ellis Horwood)

References

External links