Suppose that $g$ is a nonzero, infinite-order element of a field of characteristic $0$, and $d$ is a positive integer. Let $$ f(x):=(x-1)(x-g)\dotsb(x-g^{d-1}). $$ Suppose, furthermore, that $1<m<d/2$ is yet another integer, and $Q$ is a polynomial such that $$ (Qf)^{(k)} = P_kf + R_k,\quad 1\le k\le \deg Q $$ with some polynomials $P_k$ and $R_k$ satisfying $\deg R_k<m$. (In plain English, this means that the formal derivatives $(Qf)^{(k)}$ are divisible by $f$ up to terms of degree smaller than $m$.) How large can the degree of $Q$ be under these assumptions?
Clearly, $Q$ can be a constant polynomial; on the other hand, I can show that the assumptions imply $\deg Q\le m$. I wonder how close can $\deg Q$ get to this upper bound.
I am also interested in the situation where $(Qf)^{(k)}=P_kf+R_k$ holds only for $1\le k\le K$ with some $K<\deg Q$. Loosely speaking, in this case I want to have $K$ large, and $\deg Q$ small.
