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 Fibonacci

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:

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: