A multi-variable theta product
A clever use of the residue theorem leads the authors to the following result: \[ \prod_{j=1}^{k+1}\frac{F_k(q_{1,j},\dots,q_{j-1,j},q_{j+1,j},\dots ,q_{k+1,j};q_j)}{F_k(q_{1,j},\dots,q_{j-1,j},q_{j+1,j},\dots ,q_{k+1,j};1/q_j)}=\exp\left(\frac{\pi i}{2^k}\right), \] where \[ F_k(u_1,\dots,u_k;w)=\prod_{j_1,\dots , j_k=0}^\infty \frac{1+(-1)^{j_1+\cdots + j_k}wu_1^{2j_1+1}\cdots u_k^{2j_k+1}}{ 1-(-1)^{j_1+\cdots + j_k}wu_1^{2j_1+1}\cdots u_k^{2j_k+1}}, \] \[ q_{l,j}=\exp(\pm \pi i A_l/(2A_j)) \] (the sign \(\pm\) is chosen so that \(|q_{l,j}|<1\)), \[ q_j=i\exp(\pi A/(2A_j)), \] with \(A_1,\dots , A_{k+1}\) and \(A\) being complex numbers. The \(A_i\)'s satisfy the further conditions that they are nonzero and the quotient of any two of them is nonreal. Their result follows from the identity \[ \sum_{j=1}^{k+1}\sum_{n=-\infty}^\infty \frac{(-1)^ne^{\frac{(2n+1)\pi A}{2A_j}}}{(2n+1)\prod_{1\leq l \leq k+1, l\neq j}\cos\frac{(2n+1)\pi A_l}{2A_j}} = \frac{\pi}{2}. \] This identity is shown to yield a vast generalization of Ramanujan's identity found in Entry 15 of Chapter 14 of his second notebook.
scientific article; zbMATH DE number 1893626