Are there any known results/formulas regarding products like $$\prod_{k=1}^\infty\frac{1}{1-q^{k^2}}$$ or generally for the structure $$\prod_{k=1}^\infty\frac{1}{1-q^{p(k)}}$$ for some polynomial with $p$ with integer coefficients and $\deg p > 1$? I know that there are many formulas and product to sum identities if $p(k) =ak +b$ for some integers $a$ and $b$, but I could not find anything for other $p$. The products can be seen as generating functions of partitions, but this is not very helpful, because I could not find anything interesting for the partitions either.
So do we know nothing? Are there any references for this?
Edit: I found $$\prod_{k=1}^\infty\frac{1}{1-q^{k^2}} =\sum_{k=0}^\infty \frac{q^{k^2}}{ \prod_{j=0}^k (1 - q^{k^2})}$$ on OEIS at partitions into squares (A001156). This has a neat interpretation as ordering the partitions by the highest term.
