Modified Bernoulli Number

Author

Eric W. Weisstein
October 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/ModifiedBernoulliNumber.nb.

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

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

Series

Series[Log[(Exp[x/2]-Exp[-x/2])/(x/2)]/2,{x,0,10}]

CoefficientList[Normal[%],x^2]

Denominator[Rest[%]]

Denominator[Rest[CoefficientList[Normal[Series[Log[(Exp[x/2]-Exp[-x/2])/(x/2)]/2,{x,0,50}]],x^2]]]

Generating Function

SeriesCoefficient[(1/2)Log[((Exp[x/2]-Exp[-x/2])/((1/2)x))],{x,0,n}]

Series[(1/2)Log[((Exp[x/2]-Exp[-x/2])/((1/2)x))],{x,0,10}]

FullSimplify[(1/(4 n n! (1+n)!))(2 BernoulliB[1+n] n!-2 BernoulliB[1+n,1] n!+(BernoulliB[n]+BernoulliB[n,1]) (1+n)!),Mod[n,2]==1&&n>0]

FullSimplify[(1/(4 n n! (1+n)!))(2 BernoulliB[1+n] n!-2 BernoulliB[1+n,1] n!+(BernoulliB[n]+BernoulliB[n,1]) (1+n)!),Mod[n,2]==0&&n>0]

Table[((BernoulliB[n])/(2 (n^2) Gamma[n])),{n,10}]

SeriesCoefficient[(1/2)Log[((Exp[x/2]-Exp[-x/2])/((1/2)x))],{x,0,n}]

((BernoulliB[1+n]-BernoulliB[1+n,1])/(2 n (1+n)!))/.n->-1

(∑_(n=2)^∞)(((BernoulliB[n]) (x^n))/(2 (n^2) (Gamma[n])))+((ln(2))/2)==(1/2) ln(((exp((x/2))-exp(-(x/2)))/(x/2)))

((Log[2])/2)+Sum[((BernoulliB[n])/(2 (n^2) Gamma[n]))(x^n),{n,2,8}]

((Log[2])/2)+Sum[((BernoulliB[n])/(2 (n^2) Gamma[n]))(x^n),{n,2,∞}]

((Log[2])/2)+Sum[((BernoulliB[n])/(2 (n^2) Gamma[n]))(x^n),{n,2,∞},Assumptions->x>0]

FullSimplify[((Log[2])/2)+(1/2) (-Log[Gamma[1-((I x)/(2 π))]Gamma[1+((I x)/(2 π))]]),x>0]

FullSimplify[Log[(Exp[x/2]-Exp[-x/2])/(x/2)]/2,x>0]

FullSimplify[Subtract@@{Log[2]-(1/2) Log[x Csch[(x/2)]],(1/2) Log[((4 Sinh[(x/2)])/x)]},x>0]

(∑_(n=2,4,...)^∞)(((BernoulliB[n]) (x^n))/(2 (n^2) (Gamma[n])))+((ln(2))/2)==(1/2) ln(((exp((x/2))-exp(-(x/2)))/(x/2)))

((Log[2])/2)+Sum[((BernoulliB[n])/(2 (n^2) Gamma[n]))(x^n),{n,2,∞,2}]

FullSimplify[%,x>0]