Mahler Polynomial

Author

Eric W. Weisstein
November 23, 2008

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

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

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

Initialization

MahlerS[n_,x_]:=n!SeriesCoefficient[(E^((1-(E^t)+t) x)),{t,0,n}]

Sheffer Properies

ShefferConnection[{1&,2Sinh[#/2]&},{1&,Exp[#]-1&},x,t]

ShefferGF[{1&,#/Sqrt[1+#]&},x,t]

AssociatedGFinv[1+#-Exp[#]&,x,t]

Generating Function

Series[%,{x,0,5}]

s=SeriesTerms[%,x,t,5]

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

l=Table[MahlerS[n,x],{n,0,10}]//Factor

FindLinearRecurrence[l]

Plot

Table[MahlerS[n,x],{n,0,5}]//Factor

Plot[Evaluate[s],{x,-1,1},PlotStyle->{Black,Red,Yellow,Green,Blue,Violet},
AxesLabel->TraditionalForm/@{x,HoldForm[(Subscript[s, n])[x]]}]