# Hyperfactorial

gives the hyperfactorial function .

# Details • Mathematical function, suitable for both symbolic and numeric manipulation.
• Hyperfactorial is defined as for positive integers and is otherwise defined as .
• 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(26)

### 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(11)

Real domain of Hyperfactorial:

Complex domain:

Function range of Hyperfactorial on the contiguous portion of its domain:

Hyperfactorial is not an analytic function:

Nor is it meromorphic:

Hyperfactorial is neither non-increasing nor non-decreasing:

Hyperfactorial is not injective:

Hyperfactorial is not surjective:

Hyperfactorial is neither non-negative nor non-positive:

Hyperfactorial has both singularities and discontinuities for x-1:

Hyperfactorial is neither convex nor concave:

### 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(2)

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

Obtain Glaisher from a limit with Hyperfactorial and Exp functions:

## Properties & Relations(2)

Hyperfactorial is produced in Product:

FindSequenceFunction can recognize the Hyperfactorial sequence: