Hyperfactorial

Hyperfactorial[n]

gives the hyperfactorial function TemplateBox[{n}, Hyperfactorial].

Details

  • Mathematical function, suitable for both symbolic and numeric manipulation.
  • Hyperfactorial is defined as TemplateBox[{n}, Hyperfactorial]=product_(k=1)^nk^k for positive integers and is otherwise defined as TemplateBox[{z}, Hyperfactorial]=(z/(sqrt(2 pi)))^z exp(1/2 z (z-1)+TemplateBox[{{-, 2}, z}, PolyGamma2]).
  • The hyperfactorial function satisfies TemplateBox[{z}, Hyperfactorial]=z^z TemplateBox[{{z, -, 1}}, Hyperfactorial].
  • 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:

Hyperfactorial gives exact values for integer multiples of 1/2 and 1/4:

Find the positive minimum:

Visualization  (2)

Plot the Hyperfactorial function:

Plot the real part of TemplateBox[{z}, Hyperfactorial]:

Plot the imaginary part of TemplateBox[{z}, Hyperfactorial]:

Function Properties  (11)

Real domain of Hyperfactorial:

Complex domain:

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

Hyperfactorial threads elementwise over lists:

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:

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 :

Taylor expansion at a generic point:

Find the series expansion at Infinity:

Applications  (3)

Obtain Glaisher from a limit with Hyperfactorial and Exp functions:

The discriminant of the Hermite polynomial can be expressed in terms of the hyperfactorial:

The product of all nonzero elements of the Farey sequence for a few small orders:

Compare with a closed-form formula:

Properties & Relations  (3)

Use FullSimplify and FunctionExpand to simplify expressions involving Hyperfactorial:

Hyperfactorial is produced in Product:

FindSequenceFunction can recognize the Hyperfactorial sequence:

Neat Examples  (2)

Determinants of matrices built out of binomial coefficients:

Determinants of matrices built out of Bernstein polynomials:

Wolfram Research (2008), Hyperfactorial, Wolfram Language function, https://reference.wolfram.com/language/ref/Hyperfactorial.html.

Text

Wolfram Research (2008), Hyperfactorial, Wolfram Language function, https://reference.wolfram.com/language/ref/Hyperfactorial.html.

CMS

Wolfram Language. 2008. "Hyperfactorial." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Hyperfactorial.html.

APA

Wolfram Language. (2008). Hyperfactorial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hyperfactorial.html

BibTeX

@misc{reference.wolfram_2024_hyperfactorial, author="Wolfram Research", title="{Hyperfactorial}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Hyperfactorial.html}", note=[Accessed: 19-May-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_hyperfactorial, organization={Wolfram Research}, title={Hyperfactorial}, year={2008}, url={https://reference.wolfram.com/language/ref/Hyperfactorial.html}, note=[Accessed: 19-May-2024 ]}