Double Factorial

Author

Eric W. Weisstein
July 4, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Notation

Factorial2[n]//TraditionalForm

Enumeration

Table[n!!,{n,10}]

Factorial2/@Range[-1,2]

Table[{Product[k,{k,1,n,2}],Factorial2[n]},{n,1,5,2}]

Table[{Product[k,{k,2,n,2}],Factorial2[n]},{n,2,6,2}]

Products

Product[k,{k,n,1,-2}]

Product[k,{k,n,1,-2}]

And@@Equal@@@Table[{n!!,Product[k,{k,n,1,-2}]},{n,10}]

Product[k,{k,2-n mod 2,n,2}]

Product[k,{k,2-Mod[n,2],n,2}]

And@@Equal@@@Table[{n!!,Product[k,{k,2-Mod[n,2],n,2}]},{n,10}]

Powers of 10

v5.2

Table[Digits[(10^n)!!],{n,0,7}]//Timing

Table[Digits[(10^n)!!],{n,0,8}]//Timing

v6

Table[Digits[(10^n)!!],{n,0,7}]//Timing

Table[Digits[(10^n)!!],{n,0,8}]//Timing

...crash...

Limit

Limit[Log[10,10^n!!]/n,n->∞]

Series

v5.2

Series[x!!,{x,∞,1}]

Solve[k==2n-1,n]

Simplify[((1+k)/2)+(1/2)]

Table[((2^((1+k)/2))/(Sqrt[π]))((((Sqrt[(π/2)])))^((1+Cos[k π])/2))Gamma[1+(k/2)],{k,0,4}]

FullSimplify[((2^((1+k)/2))/(Sqrt[π]))((((Sqrt[(π/2)])))^((1+Cos[k π])/2))Gamma[1+(k/2)],k∈Integers&&k>0]

Table[(2^((1/4) (1-(((-1))^k)+2 k))) (π^((1/4) (-1+(((-1))^k)))) Gamma[1+(k/2)],{k,0,10}]

Table[n!!,{n,0,10}]

expr=Simplify[Normal[Series[Log[10,(2^((1/4) (1-(((-1))^k)+2 k))) (π^((1/4) (-1+(((-1))^k)))) Gamma[1+(k/2)]],{k,∞,2}]]/.k->10^n,n∈Integers&&n>0]

Table[1+Floor[expr],{n,0,20}]

v6

expr=Simplify[Normal[Series[Log[10,x!!],{x,∞,1}]]/.x->10^n,n∈Integers&&n>0]

FullSimplify[Cos[10^n π],n∈Integers&&n>0]

Refine[Cos[10^n π],n∈Integers&&n>0]

Table[Cos[(10^n) π],{n,20}]

Table[1+Floor[expr],{n,0,20}]

expr=Simplify[Simplify[Normal[Series[Log[10,x!!],{x,∞,1}]],x∈Integers&&x>0]/.x->10^n,n∈Integers&&n>0]

Table[1+Floor[expr],{n,0,20}]

Plots

<<Utilities`Plot`

PlotDiscontinuous[x!!,{x,-6,-4,-2,5},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,x!!},
VerticalAsymptotes->Range[-6,-2,2]]

ComplexPlot3D[z!!,{z,-2,2},PlotPoints->200,ClipFill->None,Mesh->False]//Timing

ComplexContourPlot[z!!,{z,-2,2}]//Timing

Special Cases

Negative Integrers

Table[{n,n!!},{n,-9,0}]

(-2 n-1)!!

Table[(-2n-1)!!,{n,0,10}]

(2 n+1)!!

Table[(2n+1)!!,{n,0,10}]

Table[(2n+1)!/(2^n n!),{n,0,10}]

(2 n-1)!!

Table[(2n-1)!!,{n,0,10}]

2 n!!

Table[2n!!,{n,0,10}]

Special Values

FunctionExpand[Factorial2[x]]//TraditionalForm

Series

(∑_(n=0)^∞)((x^n)/(n!!))==(1/2) (e^((x^2)/2)) ((Sqrt[2 π]) erf((x/(Sqrt[2])))+2)

Sum[((x^n)/(n!!)),{n,0,∞}]//FullSimplify

ser=SeriesCoefficient[(1/2) (E^((x^2)/2)) (2+(Sqrt[2 π]) Erf[(x/(Sqrt[2]))]),{x,0,n}]

Reduce[Mod[-1+n,2]==0&&n≥1&&Mod[n,2]==0,n,Integers]

FullSimplify[#,n∈Integers&&n>0]&/@ser

Table[ser,{n,10}]

Table[1/Factorial2[n],{n,10}]

(∑_(x=0)^∞)((x^(2 n))/((2 n)!!))==exp(((x^2)/2))

Sum[((x^(2n))/((2n)!!)),{x,0,∞}]//InputForm

Table[Sum[((x^(2n))/((2n)!!)),{n,0,100}]-Exp[x^2/2]/.x->Random[],{100}]//Chop//Union

V8

Sum[((x^(2n))/((2n)!!)),{x,0,∞}]

(∑_(x=0)^∞)((x^(2 n+1))/((2 n+1)!!))==(Sqrt[(π/2)]) erf((x/(Sqrt[2]))) exp(((x^2)/2))

Sum[((x^(2n+1))/((2n+1)!!)),{x,0,∞}]//InputForm

Table[Sum[((x^(2n+1))/((2n+1)!!)),{n,0,100}]-Exp[x^2/2]Sqrt[Pi/2]*Erf[x/Sqrt[2]]/.x->Random[],{100}]//Chop//Union

V8

Sum[((x^(2n+1))/((2n+1)!!)),{x,0,∞}]

Sums

Reciprocal Constant

Sum[1/n!!,{n,0,∞}]//FullSimplify

Square Sums

Sum[((2n-1)!!/(2n)!!)^2/(k+n),{n,0,Infinity}]

FullSimplify[Sum[((2n-1)!!/(2n)!!)^2,{n,0,Infinity}]]

Cube Sums

NSum[((2n-1)!!/(2n)!!)^3(-1)^n,{n,0,Infinity}]

(Gamma[9/8]/Gamma[5/4]/Gamma[7/8])^2//N