Nørlund Poynomial

Author

Eric W. Weisstein
October 8, 2008

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

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

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

Usage

?Norlund*

Notation

NorlundB[n,a,x]//TraditionalForm

Special Cases

(NorlundB3[n, a, 0])==(NorlundB[n, a])

NorlundB[n,a,0]

Sums

(∑_(n=0)^∞)(((t^n) (NorlundB[n, a]))/(n!))==((((t/(exp(t)-1))))^a)

Sum[NorlundB[n,a]((t^n)/(n!)),{n,0,∞}]

{Sum[NorlundB[n,a]((t^n)/(n!)),{n,0,20}],((((t/(Exp[t]-1))))^a)}/.{t->.4,a->.2}

(∑_(n=0)^∞)(((t^n) (NorlundB3[n, a, x]))/(n!))==((((t/(exp(t)-1))))^a) exp(t x)

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

{Sum[NorlundB[n,a,x]((t^n)/(n!)),{n,0,20}],Exp[x t]((((t/(Exp[t]-1))))^a)}/.{t->.4,a->.2,x->.7}

(NorlundB[k, a])==(∑_(j=0)^k)(((-1))^j) (Binomial[k+1, j+1]) (NorlundB[k, -j a])

Sum[(((-1))^j)Binomial[k+1,j+1]NorlundB[k,-j a],{j,0,k}]

Subtract@@@Flatten[Table[{NorlundB[k,a],Sum[(((-1))^j)Binomial[k+1,j+1]NorlundB[k,-j a],{j,0,k}]},{k,10},{a,1/10,2,1/10}],1]//N//Chop//Union

(((-1))^k) (Binomial[z, k]) (NorlundB[k, k-z])==(∑_(j=0)^k)(Binomial[j+k-1, k]) (NorlundB[k, j+k]) (Binomial[k-z, j+k]) (Binomial[k+z, k-j])

Sum[Binomial[k-z,k+j]Binomial[k+z,k-j]Binomial[k+j-1,k]NorlundB[k,j +k],{j,0,k}]

Subtract@@@Flatten[Table[{(((-1))^k)Binomial[z,k]NorlundB[k,k-z],Sum[Binomial[k-z,k+j]Binomial[k+z,k-j]Binomial[k+j-1,k]NorlundB[k,j +k],{j,0,k}]},{k,10},{z,1/10,2,1/10}],1]//N//Chop//Union

Identities

StirlingCarlitzS1[n_,m_]:=(((-1))^m)StirlingS1[n+1,n-m+1]
StirlingCarlitzS2[n_,m_]:=StirlingS2[m+n,n]

(((-1))^(n-m))StirlingCarlitzS1[n-1,n-m]

StirlingCarlitzS2[m,n-m]

(StirlingS1[n, n-k])==(Binomial[n-1, k]) (NorlundB[k, n])

(((-1))^k)StirlingCarlitzS1[n-1,k]

FullSimplify[StirlingS1[n,-k+n]-Binomial[n-1,k]NorlundB[k,n]]

Subtract@@@Flatten[Table[{StirlingS1[n,-k+n],Binomial[n-1,k]NorlundB[k,n]},{k,5},{n,k,5}],1]//Union

(StirlingS2[k+n, n])==(Binomial[k+n, k]) (NorlundB[k, -n])

StirlingCarlitzS2[n,k]

FullSimplify[StirlingS2[k+n,n]-Binomial[n+k,k]NorlundB[k,-n]]

Subtract@@@Flatten[Table[{StirlingS2[k+n,n],Binomial[n+k,k]NorlundB[k,-n]},{n,5},{k,5}],1]//Union

Generating Function

ExponentialGeneratingFunction[NorlundB[n,a],n,t]

SeriesCoefficient[((((t/(Exp[t]-1))))^a),{t,0,n}]//Normal

Series[((((t/(Exp[t]-1))))^a),{t,0,5}]//Normal

Together/@Sum[NorlundB[n,a]((t^n)/(n!)),{n,0,5}]

ExponentialGeneratingFunction[NorlundB[n,a,x],n,t]

SeriesCoefficient[((((t/(Exp[t]-1))))^a)Exp[x t],{t,0,n}]//Normal

Series[((((t/(Exp[t]-1))))^a)Exp[x t],{t,0,4}]//Normal

Together/@Sum[NorlundB[n,a,x]((t^n)/(n!)),{n,0,4}]

Enumeration

Flatten[Table[B[n,a,x]==NorlundB[n,a,x],{n,0,3},{a,3}]]//ColumnForm//TraditionalForm

Plots

<<Utilities`Plot`

GraphicsGrid[Partition[Table[Plot[Table[NorlundB[n,a,x],{n,0,5}],{x,-2,2},PlotStyle->{Black,Red,Yellow,Green,Blue,Purple},Evaluated->True,AxesLabel->{x,NorlundB[n,a,x]}],{a,4}],2],ImageSize->500]

Properties

D[NorlundB[n,a,x],x]

Integrate[NorlundB[n,a,x],x]

Series[NorlundB[n,a,x],{x,0,3}]//TraditionalForm