Pochhammer Symbol

Author

Eric W. Weisstein
November 7, 2008

This notebook downloaded from https://mathworld.wolfram.com/notebooks/SpecialFunctions/PochhammerSymbol.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Pochhammer limit function results and illustrations computed by
Michael Trott
December 1–2, 2003

Definition

Product[a+k,{k,0,n-1}]

TraditionalForm[%]

Enumeration

Table[{n,Pochhammer[x,n],Expand[Pochhammer[x,n]]},{n,0,4}]//TableForm//TraditionalForm

Reverse[CoefficientList[#,x]]&/@Table[Expand[Pochhammer[x,n]],{n,0,6}]

Special Cases

(Pochhammer[1, n])==n!

FullSimplify[FunctionExpand[Pochhammer[1,n]],n∈Integers&&n>0]

(Pochhammer[(1/2), n])==(2^(-n)) (2 n-1)!!

FullSimplify[FunctionExpand[Pochhammer[(1/2),n]],n∈Integers]

FunctionExpand[((Gamma[(1/2)+n])/(Sqrt[π])),n∈Integers&]

Table[((Gamma[(1/2)+n])/(Sqrt[π]))-(2^(-n)) (-1+2 n)!!,{n,-20,20}]//Union

<<Utilities`Simplify`

GammaSimplify[%20]

FullSimplify[%%]

Table[Pochhammer[(1/2),n],{n,10}]

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

Through[{Numerator,Denominator}[%]]

001147 (Formerly M3002)

000079 (Formerly M1129 )

Identities

(2^(2 n)) (Pochhammer[(x/2), n]) (Pochhammer[((x+1)/2), n])==(Pochhammer[x, 2 n])

FullSimplify[(2^(2n))Pochhammer[(x/2),n]Pochhammer[((1+x)/2),n]]

FullSimplify[%-Pochhammer[x,2n]]

(2^(2 n+1)) (Pochhammer[(x/2), n+1]) (Pochhammer[((x+1)/2), n])==(Pochhammer[x, 2 n+1])

FullSimplify[(2^(2n+1))Pochhammer[(x/2),n+1]Pochhammer[((1+x)/2),n]]

FullSimplify[((Gamma[1+2 n+x])/(Gamma[x]))-Pochhammer[x,2n+1]]

Plot

<<Utilities`Plot`

Plot[Evaluate[Table[Pochhammer[x,n],{n,0,5}]],{x,-4,4},AxesLabel->TraditionalForm/@{x,Pochhammer[x,n]},PlotStyle->{Black,Red,Yellow,Green,Blue,Purple}]

Derivative

D[Pochhammer[x,n],x]

Integrate[%,x]

%//TraditionalForm

Pochhammer[x,n]Sum[(1/(j+x)),{j,0,n-1}]//TraditionalForm

Integral

Integrate[Pochhammer[x,n],x]

Euler's Transformation

Simplify[Sum[Pochhammer[α,n]/n! a[n] z^n,{n,0,5}]]

Δ[k_,a_]:=Sum[(-1)^m Binomial[k,m] a[k-m],{m,0,k}]

FullSimplify[Series[(1-z)^(-α)Sum[Pochhammer[α,n]/n! Δ[n,a](z/(1-z))^n,{n,0,20}],{z,0,5}]]

Relationship to Rising and Falling Factorials

RisingFactorial[x_,n_]:=Pochhammer[x,n]
FallingFactorial[x_,n_]:=(-1)Pochhammer[-x,n]

Product[x-k,{k,0,n-1}]

Product[x+k,{k,0,n-1}]

Sums

(∑_(k=0)^n)(((-1))^(n-k)) (x^k) (StirlingS1[n, k])==(Pochhammer[x, n])

Sum[StirlingS1[n,k]x^k(-1)^(n-k),{k,0,n}]

Refine[%,n∈Integers]

Table[Sum[StirlingS1[n,k]x^k(-1)^(n-k),{k,0,n}],{n,0,3}]//TraditionalForm

(∑_(n=0)^∞)(((((2 x))^(2 n)) (Pochhammer[t, n]) (Pochhammer[-t, n]))/((2 n)!))==cos(2 t (sin^(-1))((Sqrt[(x^2)])))

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

-(1/2) (∑_n^∞)(((Pochhammer[t, n]) (Pochhammer[-t, n]) (((4 (sin^2)(x)))^n))/((2 n)!))==(sin^2)(t x)

-(1/2)Sum[Pochhammer[t,n]Pochhammer[-t,n]/(2n)!(4Sin[x]^2)^n,{n,∞}]

FullSimplify[%]

FullSimplify[%,0<x<π/2]

Pochhamer Limit Function

Product[x+i/k,{i,0,k}]

f[k_,x_]:=(1/k)^(1+k)Pochhammer[k x,1+k]

Asymptotic

(x^(1/2-k*x)*(1+x)^(1/2+k+k*x))/E^k

Shape of convergence

<<Utilities`Typesetting`

With[{xm=.5422114,ym=0.958833},
Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
{
ContourPlot[Re[-1+Log[(x+I y)^-(x+I y) (1+(x+I y))^(1+(x+I y))]],{x,-2,2},{y,-2,2},PlotPoints->80,Contours->{0},
ContourShading->False,FrameLabel->TraditionalForm/@{x,y},
PlotLabel->TraditionalForm[Re[-1+Log[z^-z (1+z)^(1+z)]]==0],
ContourStyle->Red,Epilog->{Red,PointSize[.04],Point[{xm,0}],Point[{-(1+xm),0}],
Point[{-1/2,#ym}]&/@{-1,1},
Text[TraditionalForm[(SubPlus[x])],{xm+.3,0}],
Text[TraditionalForm[(SubMinus[x])],{-(1+xm+.3),0}],
Text[TraditionalForm[(SubMinus[y])],{-1/2,-(ym+.3)}],
Text[TraditionalForm[(SubPlus[y])],{-1/2,ym+.3}]
}
],
With[{k=100},
Plot3D[Abs[Re[f[k,x+I y]]],{x,-2,1},{y,-1,1},PlotPoints->200,
PlotRange->{0,1},Mesh->False,
AxesLabel->TraditionalForm/@{x,y,\[Null]},PlotLabel->TraditionalForm[Abs[Re[f[x]]]],AspectRatio->Automatic
]
]
}]
]]]

Value to which it converges

1/2 (Log[z]+Log[1+z])

Maximum x

FullSimplify[Re[-1+Log[(x+I y)^-(x+I y) (1+(x+I y))^(1+(x+I y))]]/.y->0,#]&/@{x<0,x>0}

FindRoot[-1+x Log[1+(1/x)]+Log[1+x],{x,1/2},WorkingPrecision->20]

FindRoot[-1+Re[Log[(x^(-x)) (((1+x))^(1+x))]],{x,-1.1},WorkingPrecision->20]

Maximum y

FullSimplify[Re[-1+Log[(x+I y)^-(x+I y) (1+(x+I y))^(1+(x+I y))]]/.x->-1/2,#]&/@{y<0,y>0}

FindRoot[-1+Re[Log[(((-(1/2)+I y))^((1/2)-I y)) ((((1/2)+I y))^((1/2)+I y))]],{y,1/2},WorkingPrecision->20]

FindRoot[-1+Re[Log[(((-(1/2)+I y))^((1/2)-I y)) ((((1/2)+I y))^((1/2)+I y))]],{y,-1/2},WorkingPrecision->20]

Plot

With[{k=100},
Plot3D[Log[Abs[Re[f[k,x+I y]]]],{x,-2,2},{y,-2,2},PlotPoints->200,Mesh->False,
AxesLabel->TraditionalForm/@{x,y,\[Null]},PlotLabel->TraditionalForm[Log[Abs[Re[f[z]]]]]
]
]//Timing

With[{k=200},
Plot3D[Log[Abs[Re[f[k,x+I y]]]],{x,-2,2},{y,-2,2},PlotPoints->200,Mesh->False,
AxesLabel->TraditionalForm/@{x,y,\[Null]},PlotLabel->TraditionalForm[Log[Abs[Re[f[z]]]]]
]
]//Timing

With[{k=1000},
Plot3D[Log@Abs@Log[Abs[Re[f[k,x+I y]]]],{x,-2,2},{y,-2,2},PlotPoints->100,Mesh->False,
AxesLabel->TraditionalForm/@{x,y,\[Null]},PlotLabel->TraditionalForm[HoldForm[Log[Abs[Log[Abs[Re[f[x,y]]]]]]]]
]]//Timing

<<Utilities`Typesetting`

cp1=DensityPlot[Sin[Arg@f[1024,N[x+I y]]],{x,-3,2},{y,-2,2},PlotPoints->256,Mesh->False,DisplayFunction->Identity,ColorFunction->Hue,PlotRange->{-1,1}];

cp2=DensityPlot[Sin[Arg@f[2048,N[x+I y]]],{x,-3,2},{y,-2,2},PlotPoints->512,Mesh->False,DisplayFunction->Identity,ColorFunction->Hue,PlotRange->{-1,1}];

ListDensityPlot[ListConvolve[Table[1,{2},{2}]/2,cp1[[1]]],Mesh->False,ColorFunction->Hue,PlotRange->{-1,1},
FrameLabel->TraditionalForm/@{x,y},PlotLabel->TraditionalForm[Sin[Arg[f[z]]]],ImageSize->300
]//Timing

ListDensityPlot[ListConvolve[Table[1,{2},{2}]/2,cp2[[1]]],Mesh->False,ColorFunction->Hue,PlotRange->{-1,1},
FrameLabel->TraditionalForm/@{x,y},PlotLabel->TraditionalForm[Sin[Arg[f[z]]]],ImageSize->300
]//Timing