Bell Polynomial

Author

Eric W. Weisstein
September 13, 2008

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

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

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

Plot

Plot[Evaluate[Table[BellB[n,x],{n,5}]],{x,-1,1},PlotStyle->{Red,Yellow,Green,Blue,Violet}]

Special Values

(BellB2[n, 1])==(BellB[n])

BellB[n,1]

(BellB2[n, 0])==(BernoulliB[n])

BernoulliB[n,0]

Enumeration

Table[BellB[n,x],{n,0,6}]//TableForm//TraditionalForm

Reverse/@(CoefficientList[#,x]&/@Table[BellB[n,x],{n,0,5}])

Flatten[%]

Series

Series[BellB[n,x],{x,0,5}]

V5

<<DiscreteMath`RSolve`

SeriesTerm[BellB[n,x],{x,0,k}]

V7

SeriesCoefficient[BellB[n,x],{x,0,k}]

Sums

(∑_(k=0)^n)(x^k) (StirlingS2[n, k])==(BellB2[n, x])

Sum[StirlingS2[n,k]x^k,{k,0,n}]

(e^(-x)) (∑_(k=0)^∞)(((k^n) (x^k))/(k!))==(BellB2[n, x])

Exp[-x]Sum[k^n x^k/k!,{k,0,Infinity}]

x (∑_(k=1)^n)(BellB2[k-1, x]) (Binomial[n-1, k-1])==(BellB2[n, x])

x Sum[Binomial[n-1,k-1]BellB[k-1,x],{k,n}]

(∑_(k=0)^n)(({{n}, {k}})) (Subscript[B, k])(x) (Subscript[B, n-k])(y)==(Subscript[B, n])(x+y)

Sum[Binomial[n,k]BellB[k,x]BellB[n-k,y],{k,0,n}]

Derivative and Integral

Derivative

D[BellB[n,x],x]

Integrate[D[BellB[n,x],x],x]

Integrate

Integrate[BellB[n,x],x]

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

Sum[Integrate[Exp[-x](((k^n)(x^k))/(k!)),x],{k,0,n}]

FullSimplify[%]

Generating Function

s=Series[Exp[(Exp[t]-1)x],{t,0,5}]//Normal

Sum[((BellB[n,x])/(n!))(t^n),{n,0,5}]

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

ExponentialGeneratingFunction[BellB[n,x],n,t]

SeriesCoefficient[Exp[(Exp[t]-1)x],{t,0,n}]

Expand[CoefficientList[s,t]Table[n!,{n,0,5}]]//TableForm//TraditionalForm