A triple product identity for the three-parameter cubic theta function (Q2369358)

The general cubic theta function \(a(q,\zeta,z)==\sum q^ {m^ 2+mn+n^ 2}\zeta^ {m+n} z^ {m-n}\) and its variants were introduced and studied by \textit{S. Bhargava} [J. Math. Anal. Appl. 193, No. 2, 543--558 (1995; Zbl 0843.33007)] where it was shown how these functions unified and generalized several modular equations of \textit{M. D. Hirschhorn, F. Garvan}, and \textit{J. Borwein} [Can. J. Math. 45, No. 4, 673--694 (1993; Zbl 0797.33012)]. The main result of this paper is the following triple product identity \[ a(q,\zeta,z)a(q,\zeta,\omega z)a(q,\zeta,\omega^ 2 z)= b(q)[a(q)a(q^ 3,\sqrt{z^ 9/ \zeta^ 3},\sqrt{z^ 3\zeta^ 3})-c(q^ 3)a(q,z^ 3,\zeta)] \] where \[ \begin{aligned} a(q,\zeta,z) &=\sum q^ {m^ 2+mn+n^ 2}\zeta^ {m+n} z^ {m-n},\\ a(q) &=\sum q^ {m^ 2+mn+n^ 2},\\ b(q) &=\sum \omega^ {m-n}q^ {m^ 2+mn+n^ 2},\;\omega=e^ {2\pi i/3},\quad \text{and}\\ c(q) &=\sum q^ {(m+1/3)^ 2+(m+1/3)(n+1/3)+(n+1/3)^ 2}.\end{aligned} \] Finally the authors also determine a two-parameter family of zeros of \(a(q,\zeta,z)\).