Hyperfactorial

Hyperfactorial[n]

gives the hyperfactorial function .

Details

  • Mathematical function, suitable for both symbolic and numeric manipulation.
  • Hyperfactorial is defined as for positive integers .
  • Hyperfactorial is defined as for positive integers and is otherwise defined as InterpretationBox[H, Hyperfactorial, Editable -> False, Selectable -> False](z)=(z/(sqrt(2 pi)))^z exp(1/2 z (z-1)+TemplateBox[{{-, 2}, z}, PolyGamma2]).
  • The hyperfactorial function satisfies .
  • Hyperfactorial can be evaluated to arbitrary numerical precision.
  • Hyperfactorial automatically threads over lists.

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (19)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values  (4)

Values at fixed points:

Value at zero:

For a simple parameter, Hyperfactorial gives exact values:

Find the positive minimum:

Visualization  (2)

Plot the Hyperfactorial function:

Plot the real part of :

Plot the imaginary part of :

Function Properties  (4)

Real domain of Hyperfactorial:

Complex domain:

Function range of Hyperfactorial:

Hyperfactorial threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (2)

First derivative with respect to n:

Higher derivatives with respect to n:

Plot the higher derivatives with respect to n:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Taylor expansion at a generic point:

Applications  (1)

The discriminant of the Hermite polynomial is related to the hyperfactorial:

Properties & Relations  (2)

Hyperfactorial is produced in Product:

FindSequenceFunction can recognize the Hyperfactorial sequence:

Introduced in 2008
 (7.0)