I am looking for examples of sequences of polynomials $(p\_{k}(x))\_{k=1}^{\infty}$ with positive integer coefficients where $p\_{k}(0)=1$ for all $k\geq 1$ and where there is a positive integer $r$ where 
$$\frac{1}{1-rx}=\prod\_{k=1}^{\infty}p\_{k}(x)$$
whenever $x\in(0,\frac{1}{r})$.


One can easily construct many examples of polynomials $p\_{k}(x)$ that satisfy this infinite product formula using recursion. Let us therefore limit the scope of this question to sequences $(p\_{k}(x))\_{k=1}^{\infty}$ that have a closed-form expression (unless there is a very good reason to deviate) and are sequences that Srinivasa Ramanujan may be interested in.


Let me get us started with a couple of examples.


**Ex 1:** $$\frac{1}{1-x}=\prod\_{k=0}^{\infty}(1+x^{2^{k}})$$


More generally, if $a\_{k}=b\_{0}\dots b\_{k-1}$ for all $k\geq 0$, ($a\_{0}=1$ being an empty product), then
$$\frac{1}{1-x}=\prod\_{k=0}^{\infty}p\_{k}(x)$$
where $$p\_{k}(x)=\frac{x^{a\_{k+1}}-1}{x^{a\_{k}}-1}=
1+x^{a\_{k}}+x^{2a\_{k}}+\dots+x^{a\_{k+1}}.$$


**Ex 2:** $$\frac{1}{1-2x}=\prod\_{k=1}^{k}(1+x^{k})^{a(k)}$$
where $a(k)$ is sequence [A-306156](https://oeis.org/A306156) in the OEIS. Similar products exists for $\frac{1}{1-rx}$ for all $r>1$.


Polynomials that satisfy these conditions naturally arose in set theory in my investigations of rank-into-rank cardinals which float near the top of the large cardinal hierarchy.



Factoring $\frac{1}{1-rx}$ into an infinite products of polynomials
Slightly generalized from Ex. 1: 
$$ \frac{1}{1-r x} = \prod\_{j=0}^\infty \sum\_{i=0}^{m-1} (r x)^{i m^j} $$


for integers $m \ge 2$.


