A three-term identity for products of three theta functions (Q2353030)

In previous articles, the author, \textit{M. D. Monina} and \textit{M. D. Monina} (see, e.g., [Dokl. Math. 87, No. 2, 202--204 (2013); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk, Vol. 449, No. 5, 503--506 (2013; Zbl 1352.11037); Dal'nevost. Mat. Zh. 13, No. 1, 15--34 (2013; Zbl 1300.11035)]), continuing the tradition going back to Liouville, suggested new arithmetic methods for proving classical identities for Jacobi theta functions. In this paper a new arithmetic construction of this kind is proposed, which is used to prove a new identity for theta functions, namely \[ \theta_1(z;q) = -i \sum_{n=-\infty}^{\infty} (-1)^{m} e^{(2m+1)iz} q^{(m+\frac12)^2}, \] \[ \theta_3(z;q) = \sum_{n=-\infty}^{\infty} e^{2m iz} q^{(m)^2}. \] Theorem. For any complex \(w_1,w_2,w_3\) and for \(|q|<1,\) \[ \begin{multlined} \theta_3(-w_1+w_2+w_3;q^2) \theta_1(w_1;q) \theta_1(-w_2+w_3;q)\\ + \theta_3(w_1-w_2+w_3;q^2) \theta_1(w_2;q) \theta_1(w_1-w_3;q)\\ + \theta_3(w_1+w_2-w_3;q^2) \theta_1(w_3;q) \theta_1(-w_1+w_2;q) = 0. \end{multlined} \]