オイラーの分割恒等式
= ∏ m = 1 ∞ 1 − x 2 m ( 1 − x 2 m − 1 ) ( 1 − x 2 m ) = ∏ m = 1 ∞ 1 1 − x 2 m − 1 {\displaystyle {\begin{aligned}\prod _{m=1}^{\infty }\left(1+x^{m}\right)&=\prod _{m=1}^{\infty }{\frac {\left(1+x^{m}\right)\left(1-x^{m}\right)}{1-x^{m}}}\\&=\prod _{m=1}^{\infty }{\frac {1-x^{2m}}{1-x^{m}}}\\&={\frac {1-x^{2\cdot 1}}{1-x^{1}}}\cdot {\frac {1-x^{2\cdot 2}}{1-x^{2}}}\cdot {\frac {1-x^{2\cdot 3}}{1-x^{3}}}\cdot {\frac {1-x^{2\cdot 4}}{1-x^{4}}}\cdot ...\\&=\prod _{m=1}^{\infty }{\frac {1-x^{2m}}{\left(1-x^{2m-1}\right)\left(1-x^{2m}\right)}}\\&=\prod _{m=1}^{\infty }{\frac {1}{1-x^{2m-1}}}\\\end{aligned}}}