I'm looking for examples of polynomial sequences $ (p_ {k} (x)) _ {k = 1} ^ { infty} $ with positive integer coefficients where $ p_ {k} (0) = 1 $ for all $ k geq 1 $ and where is there a positive integer? $ r $ from where

$$ frac {1} {1-rx} = prod_ {k = 1} ^ { infty} p_ {k} (x) $$

whenever $ x in (0, frac {1} {r}) $,

It is easy to construct many examples of polynomials $ p_ {k} (x) $ which fulfill this infinite product formula by recursion. Let us limit the scope of this question to sequences $ (p_ {k} (x)) _ {k = 1} ^ { infty} $ which have a closed expression (unless there is a very good reason for the deviation) and are sequences in which Srinivasa Ramanujan may be interested.

Let me start with a few examples.

**Ex 1:** $$ frac {1} {1-x} = prod_ {k = 1} ^ { infty} (1 + x ^ {2 ^ {k}}) $$

**Ex 2:** $$ frac {1} {1-2x} = prod_ {k = 1} ^ {k} (1 + x ^ {k}) ^ {a (k)} $$

from where $ a (k) $ is sequence A-306156 in the OEIS. Similar products are available for $ frac {1} {1-rx} $ for all $ r> 1 $,

Polynomials that meet these conditions, of course, have arisen in set theory in my investigations of rank-in-rank cardinals hovering near the top of the great cardinal hierarchy.