Parity results for certain partition functions (Q1591448)

Let \[ (a;q)_\infty = \prod_{n=1}^\infty (1-aq^{n-1}). \] An example of the type of congruences proved in this paper is \[ \prod_{n\in S} \frac{1}{(1-q^n)} \equiv 1+\sum_{n\geq 1} q^{n^2}+\sum_{n\geq 1}q^{3n^2} \pmod{2},\tag{\(\dagger\)} \] where \[ S =\{n>0: n\equiv \pm(1,2,3,4) \text{ or } 6 \pmod{12}\}. \] From this congruence, one deduces that the number of partitions \(n\) into parts \(\pm(1,2,3,4)\) or \(6\pmod{12}\) is odd unless \(n\) is a square or three times a square. To establish the congruences such as (\(\dagger\)), the author first writes \[ \sum_{n\geq 1}q^{n^2} \equiv \sum_{n\geq 1}\frac{q^n}{1-q^n} \pmod{2} \] and observes that modulo 2, certain combinations of the series on the right hand side can be written in terms of infinite products. The proofs are then completed using addition formulas such as \[ \begin{split}\frac{1}{2}&\left\{ \frac{(-aq;q^2)_\infty(-a^{-1}q;q^2)_\infty}{(aq;q^2)_\infty(a^{-1}q;q^2)_\infty} +\frac{(-bq;q^2)_\infty(-b^{-1}q;q^2)_\infty}{(bq;q^2)_\infty(b^{-1}q;q^2)_\infty} \right\}\\ &= \frac{(abq^2;q^4)_\infty(a^{-1}b^{-1}q^2;q^4)_\infty(ab^{-1}q^2;q^4)_\infty(a^{-1}bq^2;q^4)_\infty (q^4;q^4)_\infty^2}{(aq;q^2)_\infty(a^{-1}q;q^2)_\infty(bq;q^2)_\infty(b^{-1}q;q^2)_\infty (q^2;q^2)_\infty^2}.\end{split} \] We indicate here that the above identity and its proof can also be found on p. 45 of ``Ramanujan's Notebooks, Part III'' by \textit{B. C. Berndt} (Springer, New York) (1991; Zbl 0733.11001).