Some combinatorial identities and arithmetical applications
The author describes an elementary method of raising both members of the Jacobi triple-product identity to the second and fourth powers. For example, the second power becomes \[ \prod^{\infty}_{n=1}(1- x^{2n})^ 2\quad (1+a x^{2n-1})^ 2\quad (1+a^{-1} x^{2n-1})^ 2= \] \[ =\sum^{\infty}_{-\infty}x^{2n^ 2}\sum^{\infty}_{- \infty}x^{2n^ 2} a^{2n}+x\sum^{U}_{-\infty}x^{2n(n+1)}\quad \sum^{\infty}_{-\infty}x^{2n(n+1)} a^{2n+1}. \] Some arithmetical applications are also noted.
scientific article; zbMATH DE number 3937267