Introduction to the Fibonacci and Lucas numbers

The sequence now known as Fibonacci numbers (sequence 0, 1, 1, 2, 3, 5, 8, 13...) first appeared in the work of an ancient Indian mathematician, Pingala (450 or 200 BC). Pingala's work with the mountain of cadence (now known as Pascal's triangle) made him the first known person to have looked into Fibonacci numbers. Next, another Indian mathematician, Virahanka (6th century AD), took note of the Fibonacci sequence through analysis of a completely different problem. Virahanka considered the following problem: assuming that lines of n units are composed of syllables that can be long or short—a long syllable takes twice as long as a short syllable to articulate—and each line of n units takes the same time to articulate no matter how it is composed, how many different combinations of syllables are there for each line of length n? Research of this question was continued by the Indian scholar Hemachandra and the Indian mathematician Gopala in the 12th century. Almost a half century later, the sequence was studied by the man whose name is most heavily linked to Fibonacci numbers, Leonardo of Pisa, a.k.a. Fibonacci (1202). Fibonacci considered the famous growth of an idealized rabbit population problem.

Later, European mathematicians began to study various aspects of Fibonacci numbers. Researchers included J. Kepler (1608), A. Girard (1634), R. Simpson (1753), É. Léger (1837), É. Lucas (1870, 1876–1880), G. H. Hardy, and E. M. Wright (1938). From this group, it was Francois Edouard Anatole Lucas (1870, 1876–1880) who gave Fibonacci numbers their name. He also investigated a similar sequence (sequence 2, 1, 3, 4, 7, 11, 18, 29, …), which was later coined Lucas numbers. In many works these sequences are notated and () to represent the first letters of the last names Fibonacci and Lucas. Eventually, it was established that both sequences can be analytically extended on complex -planes and that they satisfy the same three‐term recurrence relation, reflecting that the Fibonacci and Lucas numbers are the sums of two neighboring terms:

For integer arguments, Fibonacci and Lucas numbers can be elegantly represented through the symmetric relations (including the golden ratio ):

Definitions of the Fibonacci and Lucas numbers:

For any complex , the Fibonacci numbers and Lucas numbers are defined by the formulas:

where is the golden ratio .

Connections within the group of the Fibonacci and Lucas numbers and with other function groups

Representations through more general functions

The Fibonacci and Lucas numbers and have the following representations through more general functions including some hypergeometric functions and Meijer G functions:

Representations of Fibonacci and Lucas numbers through each other and through elementary functions

The Fibonacci and Lucas numbers and can be represented through each other by the following formulas:

The Fibonacci and Lucas numbers and have the following representations through elementary functions:

The best-known properties and formulas of the Fibonacci and Lucas numbers

Simple values at zero and infinity

The Fibonacci and Lucas numbers and have the following values at zero and infinity:

Specific values for specialized variables

The Fibonacci and Lucas numbers and with integer argument can be represented by the following formulas:

For the cases of integer arguments , the values of the Fibonacci and Lucas numbers and can be described by the following table:

Analyticity

The Fibonacci and Lucas numbers and are entire analytical functions of that are defined over the whole complex -plane:

Periodicity

The Fibonacci and Lucas numbers and do not have periodicity.

Parity and symmetry

The Fibonacci and Lucas numbers and generically do not have parity, but they have mirror symmetry:

Poles and essential singularities

The Fibonacci and Lucas numbers and have only the singular point . It is an essential singular point.

Branch points and branch cuts

The Fibonacci and Lucas numbers and do not have branch points and branch cuts over the complex -plane.

Series representations

The Fibonacci and Lucas numbers and have the following series expansions (which converge in the whole complex -plane):

Asymptotic series expansions

The asymptotic behavior of the Fibonacci and Lucas numbers and is described by the following formulas:

Other series representations

The Fibonacci and Lucas numbers and for integer nonnegative can be represented through the following sums involving binomials:

Integral representations

The Fibonacci and Lucas numbers and have the following integral representations on the real axis:

Generating functions

The Fibonacci and Lucas numbers and can be represented as the coefficients of the series of the corresponding generating functions:

Transformations: Addition formulas

The Fibonacci and Lucas numbers and satisfy numerous addition formulas:

Transformations: Multiple arguments

The Fibonacci and Lucas numbers and satisfy numerous identities, for example the following multiple argument formulas:

Transformations: Products and powers of the direct function

The Fibonacci and Lucas numbers and satisfy numerous identities for products and powers:

Identities

The Fibonacci and Lucas numbers and are solutions of the following simple difference equation with constant coefficients:

The Fibonacci and Lucas numbers and satisfy numerous recurrence identities:

Other identities for Fibonacci and Lucas numbers and are just functional identities:

Complex characteristics

The Fibonacci and Lucas numbers and have the following complex characteristics for complex arguments:

Differentiation

The Fibonacci and Lucas numbers and have the following representations for derivatives of the first and orders or the arbitrary fractional order :

Differential equations

The Fibonacci and Lucas numbers and satisfy the following third-order linear differential equation:

where , , and are arbitrary constants.

Indefinite integration

Some indefinite integrals for Fibonacci and Lucas numbers and can be evaluated as follows:

Laplace transforms

Laplace transforms of the Fibonacci and Lucas numbers and can be represented by the following formulas:

Summation

There exist many formulas for finite summation of Fibonacci and Lucas numbers, for example:

Here are some corresponding infinite sums:

And here are some multiple sums:

Limit operation

Some formulas including limit operations with Fibonacci and Lucas numbers and take on symmetrical forms:

Other identities

The Fibonacci numbers can be obtained from the evaluation of some determinates, for example:

Applications of the Fibonacci and Lucas numbers

Fibonacci and Lucas numbers have numerous applications throughout algebraic coding theory, linear sequential circuits, quasicrystals, phyllotaxies, biomathematics, and computer science.