In[]:=
ClearAll[BernSum];​​BernSum[n_,0]=1;​​BernSum[0,1]=0;​​BernSum[n_,1]:=
n
6
;​​BernSum[n_,m_]/;m>n:=0;​​BernSum[n_,m_]:=Sum[BernoulliB[2j1]/(2j1)!BernoulliB[2j2]/(2j2)!​​BernSum[n-1,m-j1-j2],{j1,0,m},{j2,Max[0,m-n+1-j1],m-j1}];​​A[n_,N_]:=(-1)^nN-(-1)^n2^(2n)Sum[BernoulliB[2j0]/(2j0)!BernSum[n,n-j0]N^(2j0),{j0,0,n}];
In[]:=
BernSum[n,0]
Out[]=
1
In[]:=
BernSum[n,1]
Out[]=
n
6
In[]:=
FunctionExpand[-(-1)^n2^(2n)(BernoulliB[2j0]/(2j0)!BernSum[n,1]N^(2j0))/.j0->n-1]
Out[]=
-
n
(-1)
-1+2n
2
n
2(-1+n)
N
BernoulliB[2(-1+n)]
3Gamma[-1+2n]
In[]:=
FunctionExpand[Zeta[2n]]
(-1)^(k-1)BernoulliB[2k]2^(2k-1)Pi^(2k)/(2k)!
Out[]=
-1+k
(-1)
-1+2k
2
2k
π
BernoulliB[2k]
(2k)!
In[]:=
Brule=BernoulliB[2k_]:>(2k)!
-1+k
(-1)
1-2k
2
-2k
π
Zeta[2k]
Out[]=
BernoulliB[2k_](2k)!
-1+k
(-1)
1-2k
2
-2k
π
Zeta[2k]
In[]:=
Simplify[(-(-1)^n2^(2n)(BernoulliB[2j0]/(2j0)!BernSum[n,1]q^(2j0-2n))/.j0->n-1)/.Brule]
Out[]=
-
4
2n
(-1)
n
2-2n
π
Zeta[2(-1+n)]
3
2
q