ヤコビの三重積
= a m ( x ) ∑ k = 0 ∞ b k ( x 2 ) x 2 k m ( 1 − x 2 m + 2 ) ( 1 − x 2 m + 4 ) ⋯ ( 1 − x 2 ( m + k ) ) = a m ( x ) S ( x 2 , x 2 m ) = x m 2 ( 1 − x 2 ) ( 1 − x 4 ) ⋯ ( 1 − x 2 m ) × ( 1 − x 2 m + 2 ) ( 1 − x 2 m + 4 ) ⋯ = x m 2 ( 1 − x 2 ) ( 1 − x 4 ) ⋯ = x m 2 Q ( x ) {\displaystyle {\begin{aligned}=&a_{m}(x)\sum _{k=0}^{\infty }{\frac {b_{k}(x^{2})x^{2km}}{(1-x^{2m+2})(1-x^{2m+4})\cdots (1-x^{2(m+k)})}}\\=&a_{m}(x)S(x^{2},x^{2m})\\=&{\frac {x^{m^{2}}}{(1-x^{2})(1-x^{4})\cdots (1-x^{2m})\times (1-x^{2m+2})(1-x^{2m+4})\cdots }}\\=&{\frac {x^{m^{2}}}{(1-x^{2})(1-x^{4})\cdots }}\\=&{\frac {x^{m^{2}}}{Q(x)}}\end{aligned}}}
に一致する。よって
Q ( x ) R ( x , z ) R ( x , z − 1 ) = ∑ m = − ∞ ∞ x m 2 z m {\displaystyle Q(x)R(x,z)R(x,z^{-1})=\sum _{m=-\infty }^{\infty }x^{m^{2}}z^{m}}
が成り立つ。