A note on a generalized circular summation formula of theta functions (Q413418)

The author points out an error in a formula which was stated without proof by \textit{S. H. Chan} and \textit{Z.-G. Liu} [J. Number Theory 130, No. 5, 1190--1196 (2010; Zbl 1196.33016)], and he proves a corrected version of the formula. As an application he proves transformation formulas concerning the substitution \((x,y,\tau)\mapsto ({x\over\tau}, {y\over\tau},-{1\over\tau})\) for so-called cubic theta functions, one of which is defined by NEWLINE\[NEWLINEa(x,y,\tau)= \sum^\infty_{m,n=-\infty} \exp(2\pi i(m^2+ mn+ n^2)\tau+ 2i(m(2x+ y)+ n(x+ 2y))),NEWLINE\]NEWLINE where \(x\), \(y\), \(\tau\) are complex variables with \(\tau\) in the upper half plane.