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These results have the form that if \\begin{equation*} \\frac{(q^{r-tk}, q^{mk-(r-tk)}; q^{mk})_\\infty}{(q^r,q^{mk-r}; q^{mk})_\\infty} =: \\sum_{n=0}^\\infty c_nq^n, \\end{equation*} for certain integers $k$, $m$ $s$ and $t$, where $r=sm+t$, then $c_{kn-rs}$ is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon, but also give results for cases not covered by the theorems of Alladi and Gordon. 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