Fibonacci2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n

For fixed z

General characteristics

Domain and analyticity

The function (Subscript[F, n])(z)is defined over ℕ⊗ℂ. For fixed n, the function (Subscript[F, n])(z) is a polynomial in z of degree n.

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

The function (Subscript[F, n])(z) is polynomial and has pole of order n at z =(OverTilde[∞]).

Branch points

With respect to z

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

Branch cuts

With respect to z

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

Series representations

Generalized power series

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

Expansions at z==0

Expansions at z==2i

Expansions at z==-2i

Expansions at z==∞

Integral representations

On the real axis

Of negative order

Generating functions

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Differentiation

Low-order differentiation

With respect to z

Symbolic differentiation

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Summation

Multiple sums

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1

Involving HypergeometricPFQ

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With elementary functions

Zeros

History

– M. Bicknell (1970)

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