Rising Factorial

Author

Eric W. Weisstein
August 12, 2003

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Definition

(Subscript[(x), n])=((Γ(n+x))/(Γ(x)))

FullSimplify[Pochhammer[x,n]-Gamma[x+n]/Gamma[x]]

Identities

FunctionExpand[{Pochhammer[1,n],n!}]

Enumeration

Pochhammer[x,0]

Table[Pochhammer[x,n],{n,0,5}]//Expand//ColumnForm

Plots

MakeBoxes[RisingFactorial[x_,n_],TraditionalForm]:=SuperscriptBox[MakeBoxes[x,TraditionalForm],RowBox[{"(",MakeBoxes[n,TraditionalForm],")"}]]

<<Graphics`Colors`

Plot[Evaluate[Table[Pochhammer[x,n],{n,0,5}]],{x,-2,2},PlotStyle->{Black,Red,Yellow,Green,Blue,Violet},
AxesLabel->TraditionalForm/@{x,RisingFactorial[x,n]}]

Derivative

(D[Pochhammer[x,n],x]/.Pochhammer->RisingFactorial)//TraditionalForm

Integral

Integrate[Pochhammer[x,n],x]