Subfactorial

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

The numbers Subfactorial(n) represent the number of complete permutations (i.e., derangements) of n objects (that is the number of a permutation in which no object appears in its "natural" (i.e., ordered) place).

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

!z is an analytical function of z which is defined in the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function !z has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

Branch points

The function !z does not have branch points.

Branch cuts

The function !z does not have branch cuts.

Series representations

Generalized power series

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

Asymptotic series expansions

Integral representations

On the real axis

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

With respect to a

Integration

Indefinite integration

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric1FRegularized

Involving Hypergeometric1F1

Involving hypergeometric HypergeometricU

Through other functions

Involving some hypergeometric-type functions

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