The Fibonacci and Lucas numbers and have the following values at zero and infinity:
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:
The Fibonacci and Lucas numbers and are entire analytical functions of that are defined over the whole complex plane:
The Fibonacci and Lucas numbers and do not have periodicity.
The Fibonacci and Lucas numbers and generically do not have parity, but they have mirror symmetry:
The Fibonacci and Lucas numbers and have only the singular point . It is an essential singular point.
The Fibonacci and Lucas numbers and do not have branch points and branch cuts over the complex plane.
The Fibonacci and Lucas numbers and have the following series expansions (which converge in the whole complex plane):
The asymptotic behavior of the Fibonacci and Lucas numbers and is described by the following formulas:
The Fibonacci and Lucas numbers and for integer nonnegative can be represented through the following sums involving binomials:
The Fibonacci and Lucas numbers and have the following integral representations on the real axis:
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:
The Fibonacci and Lucas numbers and satisfy numerous identities, for example the following multiple argument formulas:
The Fibonacci and Lucas numbers and satisfy numerous identities for products and powers:
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:
The Fibonacci and Lucas numbers and have the following complex characteristics for complex arguments:
The Fibonacci and Lucas numbers and have the following representations for derivatives of the first and orders or the arbitrary fractional order :
The Fibonacci and Lucas numbers and satisfy the following thirdorder linear differential equation:
where , , and are arbitrary constants.
Some indefinite integrals for Fibonacci and Lucas numbers and can be evaluated as follows:
Laplace transforms of the Fibonacci and Lucas numbers and can be represented by the following formulas:
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:
Some formulas including limit operations with Fibonacci and Lucas numbers and take on symmetrical forms:
The Fibonacci numbers can be obtained from the evaluation of some determinates, for example:
