On a new circular summation of theta functions (Q971847)

An example of circular summation of theta functions is the formula \[ \sum^{n-1}_{k=0} \theta_3\Biggl(z+{k\pi\over n}, \tau\Biggr)= n\theta_3(nz, n\tau), \] proved by \textit{M. Boon}, \textit{M. L. Glasser}, \textit{J. Zak} and \textit{I. J. Zucker} [J. Phys. A 15, 3439--3440 (1982; Zbl 0543.33004)], where \[ \theta_3(z,\tau) =\sum^\infty_{n=-\infty} q^{n^2/2} e^{2niz} \] with \(q= e^{2\pi ir}\), \(\text{Im}(z)>0\) is Jacobi's theta function. The first instance of circular summation is known from Ramanujan's Lost Notebook. The authors prove formulas which comprise as special cases most of the previously known formulas on circular summation. One of them reads \[ \sum^{mn-1}_{k=0}\;\pi^n_{j=1} \theta_3\Biggl(z+ y_j+ {k\pi\over mn},\tau\Biggr)= G_{m,n}(y_1,\dots, y_n,\tau) \theta_3(mnz, m^2n\tau), \] where \(y_1,\dots, y_n\in\mathbb{C}\), \(y_1+\cdots+ y_n= 0\) and \(G_{m,n}\) is explicitly given in terms of an infinite series independent from \(z\). One among several corollaries is \[ {\eta^3(\tau)\over \eta(3\tau)}= \sum_{x,y\in\mathbb{Z}} q^{x^2+ xy+ y^2} \omega^{x-y} \] with \(\omega= e^{2\pi i/3}\), relating a quotient of Dedekind's eta function \(\eta(\tau)\) to the ring of Eisenstein integers.