20/09/18 18:33:40.22 sSB3QbM0.net
>>653
x = e^(-π) とおくと
(左辺) = x/(1+x) + 3(x^3)/(1+x^3) + 5(x^5)/(1+x^5) + ･････
= x {1/(1+x) + (3x^2)/(1+x^3) + (5x^4)/(1+x^5) + ･････ }
= x (d/dx) log[(1+x)(1+x^3)(1+x^5)････]
= x (d/dx){ log[(1+x)(1-x^2)(1+x^3)(1-x^4)(1+x^5)････]
- log[(1-x^2)(1-x^4)(1-x^6)････] }
= x (d/dx){ -log(G(-x)) + log(G(x^2)) },
ここに
G(x) = Σ[n=0,∞] p(n)･x^n,
は分割数p(n)の生成関数。
1/G(x) = (1-x)(1-x^2)(1-x^3)･･･
= Σ[m=-∞,∞] (-1)^m x^{m(3m-1)/2},