Euler Polynomial

Author

Eric W. Weisstein
December 26, 2008

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

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

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

Enumeration

Table[HoldForm[EulerE[n,x]]==EulerE[n,x],{n,0,10}]//TableForm//TraditionalForm

Plots

Plot[Evaluate[Table[EulerE[n,x],{n,5}]],{x,-1,2},PlotStyle->{Red,Yellow,Green,Blue,Violet}]

Identities

(Subscript[E, n])(x+1)+(Subscript[E, n])(x)==2 (x^n)

V5

FullSimplify[EulerE[n,x+1]+EulerE[n,x],n∈Integers]

Expand[Table[EulerE[n,x+1]+EulerE[n,x]-2x^n,{n,0,6}]]

V7

FullSimplify[EulerE[n,x+1]+EulerE[n,x],n∈Integers&&n≥0]

(((2^n) ((Subscript[B, n])(((x+1)/2))-(Subscript[B, n])((x/2))))/n)==(Subscript[E, n-1])(x)

FullSimplify[((2^n)/n)(BernoulliB[n,((x+1)/2)]-BernoulliB[n,(x/2)]),n∈Integers&&n>0]

FullSimplify[((2^n)/n)(BernoulliB[n,((x+1)/2)]-BernoulliB[n,(x/2)])-EulerE[n-1,x],n∈Integers&&n>0]

Table[(((2^n) (-BernoulliB[n,(x/2)]+BernoulliB[n,((1+x)/2)]))/n)-EulerE[-1+n,x],{n,10}]//FullSimplify

((2 ((Subscript[B, n])(x)-(2^n) (Subscript[B, n])((x/2))))/n)==(Subscript[E, n-1])(x)

FullSimplify[(2/n)(BernoulliB[n,x]-(2^n)BernoulliB[n,(x/2)]),n∈Integers&&n>0]

FullSimplify[(2/n)(BernoulliB[n,x]-(2^n)BernoulliB[n,(x/2)])-EulerE[n-1,x],n∈Integers&&n>0]

(2^n) (Subscript[E, n])((1/2))==(Subscript[E, n])

2^n EulerE[n,1/2]

(EulerE2[n, 0])==-((2 ((2^(n+1))-1) (BernoulliB[n+1]))/(n+1))

FunctionExpand[EulerE[n,0],n≥0&&n∈Integers]

FullSimplify[EulerE[n,0]+2BernoulliB[n+1]((2^(n+1))-1)/(n+1),n∈Integers&&n>=0]

Sums

(2/(Binomial[n, 2]))(∑_(k=0)^(n-2))(Binomial[n, k]) (((2^(n-k))-1) (Subscript[B, n-k]) (Subscript[B, k])(x))==(Subscript[E, n-2])(x)

2((Binomial[n,2])^(-1))Sum[Binomial[n,k](((2^(n-k))-1)BernoulliB[n-k]BernoulliB[k,x]),{k,0,n-2}]//InputForm

Table[(4*Sum[(-1+2^(-k+n))*BernoulliB[-k+n]*BernoulliB[k,x]*Binomial[n,k],{k,0,-2+n}])/((-1+n)*n)-EulerE[n-2,x],{n,2,10}]//Simplify

(2^(-n)) (∑_j^n)(((-1))^(j+n+1)) (j^n)(∑_(k=0)^(n-j))(Binomial[n+1, k])==(Subscript[E, n])(0)

Table[EulerE[n,0],{n,10}]

V5

2^(-k) Sum[(-1)^(i+1+k) i^k Sum[Binomial[k+1,L],{L,0,k-i}],{i,1,k}]

Table[2^(-k) Sum[(-1)^(i+1+k) i^k Sum[Binomial[k+1,L],{L,0,k-i}],{i,1,k}],{k,10}]

V7

Sum[Binomial[n+1,k],{k,0,n-j}]//FullSimplify

(2^(-n))Sum[(((-1))^(j+n+1))(j^n)((2^(1+n))-Binomial[1+n,1-j+n] Hypergeometric2F1[-j,1,2-j+n,-1]),{j,n}]//InputForm

Table[Sum[(-1)^(1+j+n)*j^n*(2^(1+n)-Binomial[1+n,1-j+n]*Hypergeometric2F1[-j,1,2-j+n,-1]),{j,n}]/2^n,{n,10}]

Table[EulerE[n,0],{n,10}]

(∑_(j=0)^n)(∑_(k=j)^n)(2^(-j)) (Binomial[-1, j]) (Subscript[(k), j]) (Subscript[(x), k-j]) (Subscript[𝒮, n]^((k)))≠(Subscript[E, n])(x)

Sum[Binomial[-1,j](2^(-j))Pochhammer[k,j]StirlingS2[n,k]Pochhammer[x,k-j],{j,0,n},{k,j,n}]//InputForm

Table[Sum[(Binomial[-1,j]*Pochhammer[k,j]*Pochhammer[x,-j+k]*StirlingS2[n,k])/2^j,{j,0,n},{k,j,n}]-EulerE[n,x],{n,10}]//Simplify

(∑_(k=0)^n)(Binomial[n, k]) (Subscript[E, k])(z) (Subscript[E, n-k])(w)==2 ((-w-z+1) (Subscript[E, n])(w+z)+(Subscript[E, n+1])(w+z))

Sum[Binomial[n,k]EulerE[k,z]EulerE[n-k,w],{k,0,n}]

Generating Function

Expand[CoefficientList[Series[2Exp[x t]/(Exp[t]+1),{t,0,10}],t]*Table[n!,{n,0,10}]]//TableForm

Table[EulerE[n,x],{n,0,10}]//TableForm

Relation to Sech

Table[2^k EulerE[k,1/2],{k,10}]

Series[Sech[x],{x,0,10}]

Sum[EulerE[k]x^k/k!,{k,0,10}]

sech=Series[Sech[x]-1,{x,0,10}]

c=CoefficientList[sech,x];
c*Table[n!,{n,0,10}]

sec=Series[Sec[x]-1,{x,0,10}]

c=CoefficientList[sec,x];
c*Table[(n-1)!,{n,1,Length[c]}]

b[k_]:= -k e2[k-1]/(2(2^k-1))
{Table[b[k],{k,10}],Table[BernoulliB[k],{k,10}]}//TableForm

e2[0]