Hyperfactorial

Author

Eric W. Weisstein
June 29, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Hyperfactorial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Hyperfactorial.html.

©2011 Wolfram Research, Inc. except for portions noted otherwise

Notation

V6

SpecialFunctions`Hyperfactorial[n]//TraditionalForm

V7

Hyperfactorial[n]//TraditionalForm

Properties

Not Entire

?Reduce`HolomorphicFunctionQ

Reduce`HolomorphicFunctionQ[Hyperfactorial[z],n,{-Infinity(I+1),Infinity(I+1)}]

Reduce`HolomorphicFunctionQ[Hyperfactorial[z],{-Infinity(I+1),Infinity(I+1)}]

Products

(∏_k^n)(k^k)==H(n)

Table[Product[(k^k),{k,n}],{n,10}]

V6

Product[(k^k),{k,n}]

SpecialFunctions`Hyperfactorial[n]//FunctionExpand

Table[SpecialFunctions`Hyperfactorial[n],{n,10}]

V7

Product[(k^k),{k,n}]

(∏_(k=0)^n)(Binomial[n, k])==((H(n))^2) ((Gamma[n+1])^(-(n+1)))

Product[Binomial[n,k],{k,0,n}]

Table[Product[Binomial[n,k],{k,0,n}],{n,0,10}]

Table[Hyperfactorial[n]^2 Gamma[1+n]^-(n+1),{n,0,10}]

Plots

<<Utilities`Typesetting`
<<Utilities`Plot`

Real Line

V7

g1=Plot[Through[{Re,Im}[Hyperfactorial[n]]],{n,-3,2},AxesLabel->{n,H[n]},Evaluated->True]

g2=Plot[Log[Hyperfactorial[n]],{n,0,10},AxesLabel->{n,Log[Hyperfactorial[n]]},AxesOrigin->{0,0},Method->{"AxesInFront"->True}]

...can't do LogPlot because of bug saving ticks with exponents as EPS...

GraphicsGrid[{{g1,g2}},ImageSize->400]

Complex Plane

Re/Im Plot

ComplexPlot3D[Hyperfactorial[z],{z,-3,3}]

Contour Plot

ComplexContourPlot[Hyperfactorial[z],{z,-3,3}]

ComplexContourPlot[Hyperfactorial[z],{z,-3,3},Contours->8,"MeshRegions"->False,"PackGraphicsComplex"->True]

Minimum

Plot3D[Abs[Hyperfactorial[x+I y]],{x,-2,2},{y,-2,2}]

FunctionExpand[Abs[Hyperfactorial[I y]],y∈Reals]

Plot[Abs[Hyperfactorial[I y]],{y,-10,10}]

Plot[Through[{Re,Im}[Hyperfactorial[n]]],{n,-5,2},AxesLabel->{n,H[n]},Evaluated->True]

Limits

Limit[Hyperfactorial[I y],y-># ∞]&/@{-1,1}

N[Hyperfactorial[12#I ]&/@{-1,1},20]

Identities

Table[Hyperfactorial[n],{n,5}]

H(z)==(((2 π))^((1/2) (-z-1))) (e^((ψ^((-2)))(z+1)+(1/2) z (z+1)))

Exp[(z-1)*z/2+PolyGamma[-2,z]]*(2 Pi)^(-z/2)/.z->z+1

Table[(E^((1/2) z (1+z)+PolyGamma[-2,1+z])) (((2 π))^((1/2) (-1-z))),{z,-5,5}]//FullSimplify

Table[Hyperfactorial[z],{z,-5,5}]

FullSimplify[(E^((1/2) z (1+z)+PolyGamma[-2,1+z])) (((2 π))^((1/2) (-1-z)))]

FullSimplify[(E^((1/2) z (1+z)+PolyGamma[-2,1+z])) (((2 π))^((1/2) (-1-z)))-Hyperfactorial[z]]

Subtract@@@Table[{Hyperfactorial[z],(E^((1/2) z (1+z)+PolyGamma[-2,1+z])) (((2 π))^((1/2) (-1-z)))}/.z->RandomComplex[5(I+1){-1,1}],{100}]//Chop//Union

H(n)==A (e^(Derivative[1,0][ζ](-1,n+1)-(1/12)))

FullSimplify[(E^(-(1/12)+Derivative[1,0][Zeta][-1,1+n])) Glaisher]//InputForm

Table[(E^(-(1/12)+Derivative[1,0][Zeta][-1,1+n])) Glaisher,{n,10}]//FunctionExpand

Subtract@@@Table[{(E^(-(1/12)+Derivative[1,0][Zeta][-1,1+n])) Glaisher,Hyperfactorial[n]}/.n->RandomComplex[(I+1){-1,1},WorkingPrecision->50],{100}]//Chop//Union

H(n)==exp(ζ'(-1,z+1)-ζ'(-1))

Exp[Derivative[1,0][Zeta][-1,z+1]-Zeta'[-1]]

FullSimplify[%]

Table[(E^(-(1/12)+Derivative[1,0][Zeta][-1,1+z])) Glaisher,{z,5}]//FunctionExpand

H(n)==(((2 π))^(-(z/2))) exp((∫_0^z)ln(t!)dt+(Binomial[z+1, 2]))

K2[z_]:=Module[{t},
(2Pi)^(-(z-1)/2)Exp[Binomial[z,2]+Integrate[Log[t!],{t,0,z-1}]]
]

v4.2

Table[K2[n],{n,5}]

FullSimplify[%]

N[%]

v5.0

Table[K2[n],{n,5}]

FullSimplify[%]

N[%]

V7

(((2π))^(-z/2))Exp[Binomial[z+1,2]+Integrate[Log[t!],{t,0,z}]]

(((2π))^(-z/2))Exp[Binomial[z+1,2]+Integrate[Log[t!],{t,0,z},Assumptions->z>0]]

FullSimplify[%]

Table[(E^((1/2) z (1+z)+PolyGamma[-2,1+z])) (((2 π))^((1/2) (-1-z))),{z,5}]//FunctionExpand//FullSimplify

Series

Series[Hyperfactorial[z],{z,0,3}]//FullSimplify

Series[Hyperfactorial[z],{z,∞,6}]//FullSimplify

Normal[Series[Hyperfactorial[z],{z,∞,6}]]/.z->5.

Hyperfactorial[5.]

SeriesCoefficient[Hyperfactorial[z]/((E^(-((z^2)/4))) (z^((1/12)+(1/2) z (1+z))) Glaisher),{z,∞,n}]

Series[Hyperfactorial[z]/((E^(-((z^2)/4))) (z^((1/12)+(1/2) z (1+z))) Glaisher),{z,∞,10}]//Simplify

CoefficientList[Normal[%],1/z^2]

Through[{Numerator,Denominator}[%]]

Derivatives and Integrals

Derivative

D[Hyperfactorial[x],x]

TraditionalForm[%]

Integrate[D[Hyperfactorial[x],x],x]

Integral

Integrate[Hyperfactorial[x],x]

Special Values

H(-(1/2))==((A^(3/2))/((2^(1/24)) (e^(1/8))))

V6

PowerExpand[{Exp[-Log[2]/3-12Zeta'[-1]],
2^(2/3)Pi Exp[EulerGamma-1-Zeta'[2]/Zeta[2]]}^(1/8)]

N[%]

N[KFunctionN[1/2],20]

V7

FunctionExpand[Exp[-Log[2]/3-12Zeta'[-1]]]//FullSimplify

FunctionExpand[2^(2/3)Pi Exp[EulerGamma-1-Zeta'[2]/Zeta[2]]]

((Hyperfactorial[-1/2])^8)

Hyperfactorial[-1/2]