The bestknown 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 thirdorder 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:
