Cassini formula for Fibonacci numbers

Giovanni
Cassini. Born: 8 June 1625 in Perinaldo, Republic of Genoa
(now Italy). Died: 14 Sept 1712 in Paris, France.We know
little of his parents but certainly his father was a Tuscan.
In fact Giovanni was brought up, not by his parents but by an
uncle, a brother of his mother Julia Crovesi. After spending
two years being educated at Vallebone, Cassini entered the
Jesuit College at Genoa where he studied under Casselli. After
this he studied at the abbey of San Fructuoso. He showed great
intellectual curiosity and was especially interested in
poetry, mathematics and astronomy. His first interest,
however, was in astrology rather than astronomy. He read
widely on this topic and soon was very knowledgeable, yet was
convinced that there was no truth in astrological predictions.
It was, rather strangely, his extensive knowledge of astrology
that led to his first appointment. In 1644 the Marquis
Cornelio Malvasia, who was a senator from Bologna with a great
interest in astrology, invited Cassini to Bologna. He offered
him a position in the Panzano Observatory which he was
constructing at that time. Cassini was an astronomer at the
Panzano Observatory, from 1648 to 1669. He was a professor of
astronomy at the University of Bologna. Cassini's brilliant
discoveries gave him an international reputation and led to
him being invited to Paris by Louis XIV in 1668.

He thoroughly adopted his
new country, to the extent that he became interchangeably known as
JeanDominique Cassini. Cassini became,in 1671, director of the
Paris Observatory.Cassini would remain the director of it for the
rest of his career until his death in 1712. Along with Hooke,
Cassini is given credit for the discovery of the Great Red Spot
(~1665). Cassini was the first to observe four of Saturn's moons; he
also discovered the Cassini Division (1675). Around 1690, Cassini
was the first to observe differential rotation within Jupiter's
atmosphere. In 1672 he sent his colleague Jean Richer to Cayenne,
French Guiana, while he himself stayed in Paris. The two made
simultaneous observations of Mars and thus found its parallax to
determine its distance, thus measuring for the first time the true
dimensions of the solar system. Cassini was the first to make
successful measurements of longitude by the method suggested by
Galileo, using eclipses of the satellites of Jupiter as a clock. In
1680 he studied the Cassinian curve which is the locus of a point
the product of whose distances from two fixed foci is constant. He
worked on this as part of a study of the relative motions of the
Earth and the sun and proposed this as the curve for planetary
orbits rather than the ellipse as proposed by Kepler.
The original Cassini formula
for Fibonacci numbers is :
Here, we shall give a proof
for the Giovanni Cassini`s formula:
Wellknown
Binet`s formula for Lucas numbers that can be used to prove Cassini
formula is:
from which we get Binet`s operator equation
for Fibonacci and Lucas numbers:
We shell use the next Fibonacci formula ( Vajda11,Dunlap7, Lucas1876):
However, following the operator`s calculus, we obtain
the Cassini formula for Fibonacci numbers:
Many of The Fibonacci and Lucas sequences
proprties follow immediately from the recursive rule that each term
is the sum of the two preceding terms. Similarly, it is easy to
establish corresponding form for Cassini Lucas numbers formula.
Here, we shall give a proof for the Cassini`s Lucas formula by
using general Fibonacci and Lucas Numbers relation ( Vajda20b ):
Using the context of Difference operators and
the Fibonacci space, we can derive useful results.
