html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Fibonacci

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

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: