【１】ベルヌーイ数の定義

x/(exp(x)-1)=Σ(0,∞)Bn・x^n/n!

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1=(exp(x)-1)/x・Σ(0,∞)Bn・x^n/n!

1=Σ(0,∞)x^m/(m+1)!・Σ(0,∞)Bn・x^n/n!

1=Σ(0,∞)Σ(0,∞)Bn・x^(m+n)/(m+1)!n!

k=m+n

Σm(0,∞)Σn(0,∞)=Σk(0,∞)Σn(0,k)

1=Σk(0,∞){Σn(0,k)Bn/(k+1-n)!n!}・x^k

B0＝1,Σn(0,k)Bn/(k+1-n)!n!=0

B0＝1,Σ(0,k)(k+1,n)Bn=0・・・漸化式

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(2,0)B0+(2,1)B1=0→B1=-1/2

(3,0)B0+(3,1)B1+(3,2)B2=0→B2=1/6

(4,0)B0+(4,1)B1+(4,2)B2+(4,3)B3=0→B3=0

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B1=-1/2,B2m+1=0

x/(exp(x)-1)=Σ(0,∞)Bn・x^n/n!=-x/2+Σ(0,∞)B2m・x^2m/(2m)!

xcotx=Σ(0,∞)(-1)^mB2m・(2x)^2m/(2m)!

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Σ(0,n)(n+1,k)Bk=0

Σ(0,m)(2m+1,2k)2^2k・B2k=2m+1

xtanx=Σ(0,∞)(1-2^2m)(-1)^mB2m(2x)^2m/(2m)!

Σ(1,m)(2m,2k)2^2k(1-2^2k)・B2k=-2m

x/sinx=1+2Σ(0,∞)(1-2^2m-1)(-1)^mB2m(x)^2m/(2m)!

x/(exp(x)+1)=x/2+Σ(1,∞)(1-2^2m)B2m・x^2m/(2m)!

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