【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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【２】オイラー数の定義

1/cosx=Σ(0,∞)β2m・x^2m/(2m)!

E2m=(-1)^mβ2m

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1=cosx・Σ(0,∞)β2m・x^2m/(2m)!

1=Σ(0,∞)(-1)^m/(2m)!・x^2m・Σ(0,∞)β2k・x^2k/(2k)!

1=Σ(0,∞)Σ(0,∞)(-1)^mβ2k・x^(2m+2k)/(2m)!(2k)!

n=m+k

1=Σn(0,∞){Σk(0,n)(-1)^(n-k)β2k/(2n-2k)!(2k)!}・x^2n

β0＝1,Σk(0,n)(-)^(n-k)(2n,2k)β2k=0・・・漸化式

β1=1,β4=5,β6=61,・・・

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