Fibonacci

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

(Subscript[F, ν]) is an entire analytical function of ν which is defined over the whole complex ν-plane.

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function (Subscript[F, ν]) has only one singular point at ν = (OverTilde[∞]).  It is an essential singular point.

Branch points

The function (Subscript[F, ν]) does not have branch points.

Branch cuts

The function (Subscript[F, ν]) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point ν==(Subscript[ν, 0])

Expansions at ν==0

Asymptotic series expansions

Other series representations

Integral representations

On the real axis

Of the direct function

Limit representations

Generating functions

Differential equations

Ordinary linear differential equations and wronskians

Transformations

Addition formulas

Multiple arguments

Products, sums, and powers of the direct function

Products of the direct function

Powers of the direct function

Related transformations

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Functional equations

Relations of special kind

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Integral transforms

Laplace transforms

Summation

Finite summation

Infinite summation

Multiple sums

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Generalized cases for the direct function itself

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With elementary functions

With Lucas numbers

Other identities

Identities involving determinants

Theorems

Zeckendorf theorem

Every positive integer can be decomposed in a unique way as a sum of Fibonacci numbers, such that no two of these numbers are consecutive in the Fibonacci sequence.

Fibonacci substitution

After acting on A n times with the Fibonacci substitution {A->A B,B->A} the resulting sequence contains (Subscript[F, n+1]) As and (Subscript[F, n]) Bs .

A transcendental number

(∑_(n=1)^∞)(Subscript[F, n]^(-2)) is a transcendental number.

The numbers of primary and secondary spirals in the positions of leaves

The numbers of primary and secondary spirals in the positions of leaves or scales along a plant stem are nearly always two consecutive Fibonacci numbers.

Hirmer's conjecture

The number of the largest set of nonintersecting circles arranged along the circumference of a given circle and angle 2π (1-GoldenRatio) between consecutive midpoints is given by the Fibonacci numbers (Subscript[F, n]).

History

– J. Kepler (1608)
– A. Girard (1634); R. Simpson (1753)
– É. Léger (1837)
– É. Lucas (1870, 1876–1880)
– G.H. Hardy and E.M. Wright (1938)

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