Let $ r $ to be an integer with $ r> $ 1. Assume that $ p_ {k} (x) $ is a polynomial with positive integer coefficients with $ p_ {k} (0) = $ 1 but where $ p_ {k} neq 1 $ for everyone $ k geq 0 $.

Assume that

$$ prod_ {k = 0} ^ { infty} p_ {k} (x) = frac {1} {1-rx} $$

For each $ n> $ 0, let $ t_ {n} $ to be the numerical indices k $ or $ deg (p_ {k} (x)) = n $. So is it possible to select polynomials $ (p_ {k}) _ {k geq 0} $ or

$$ | t_ {n} – frac {r ^ {n}} {n} | = O ( alpha ^ {n}) $$

for each $ alpha> $ 1?

How long can the function $ n mapsto | t_ {n} – frac {r ^ {n}} {n} | $ grow? How long can the function $ n mapsto max (0, frac {r ^ {n}} {n} -t_ {n} $ grow? For example, can we have $ max (0, frac {r ^ {n}} {n} -t_ {n}) = O ( alpha ^ {n}) $ for everyone $ alpha> $ 1?

This question is motivated by very great cardinals.