Fibonacci2General

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν

For fixed z

General characteristics

Domain and analyticity

(Subscript[F, ν])(z) is an analytical function of ν and z which is defined over (ℂ^2). For integer ν, (Subscript[F, ν])(z) degenerates to a polynomial in z. For fixed z, it is an entire function of ν.

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to ν

For fixed z, the function (Subscript[F, ν])(z) has one singular point at ν = (OverTilde[∞]).  It is an essential singular point.

Branch points

With respect to z

With respect to ν

For fixed z, the function (Subscript[F, ν])(z) does not have branch points.

Branch cuts

With respect to z

With respect to ν

For fixed z, the function (Subscript[F, ν])(z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions at ν==0

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

Expansions at z==0

Expansions at z==2i

Expansions at z==-2i

Expansions at z==∞

Asymptotic series expansions

Residue representations

Integral representations

Contour integral representations

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 ν

With respect to z

Symbolic differentiation

With respect to ν

With respect to z

Fractional integro-differentiation

With respect to ν

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving only one direct function with respect to ν

Involving one direct function and elementary functions with respect to ν

Integral transforms

Laplace transforms

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

History

– M. Bicknell (1970)

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