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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 third-order 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:
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