Stirling Polynomial

Author

Eric W. Weisstein
December 30, 2008

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

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

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

Generating Function

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

Range[0,5]!CoefficientList[Series[((((t/(1-Exp[-t]))))^(x+1)),{t,0,5}],t]//Factor

Relation to Stirling Numbers

(Subscript[S, n])(m)==(((((-1))^n) )/(Binomial[m, n]))(StirlingS1[m+1, m-n+1])

Table[{n!SeriesCoefficient[((((t/(1-Exp[-t]))))^(m+1)),{t,0,n}],((((-1))^n)/(Binomial[m,n]))StirlingS1[m+1,m-n+1]},{m,5},{n,m}]

(Subscript[S, n])[m]

Wrong in Roman, p. 129

Table[{n!SeriesCoefficient[((((t/(1-Exp[-t]))))^(m+1)),{t,0,n}],(((((-1))^n)n!)/((n-m-1)!))StirlingS2[n-m-1,-m-1]},{m,-5,-1},{n,4}]

Table[{n!SeriesCoefficient[(((((Exp[t]-1)/t)))^(-m-1)),{t,0,n}],(((((-1))^n)n!)/((n-m-1)!))StirlingS2[n-m-1,-m-1]},{m,-5,-1},{n,4}]