A new look at Hecke's indefinite theta series (Q2782416)

The author investigates the \(q\)-series of the form NEWLINE\[NEWLINE\sum_{m\geq 0,\;n\geq 0} f(m,n) q^{Q(m,n)}- \sum_{m<0,\;n<0} f(m,n) q^{Q(m,n)},NEWLINE\]NEWLINE where \(Q\) is an indefinite quadratic form on \(\mathbb{Z}^2\) and \(f(m,n)\) is a double periodic function on \(\mathbb{Z}^2\) such that the sums of \(f(m,n) q^{Q(m,n)}\) over all vertical and all horizontal lines in \(\mathbb{Z}^2\) vanish. It is proved that these series generate the same space of weight 1 modular forms as Hecke's indefinite theta series.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00024].