Classical division theorem for rational-coefficient E-functions

Let $f(X)=\sum_{n\ge0}a_nX^n/n!$ have rational normalized coefficients satisfying the E-function exponential size and common-denominator bounds. Suppose $f(1)=0$ and $f$ is annihilated by a nonzero rational polynomial differential operator $\sum_{k=0}^m p_k(X)D^k$, with $p_m\ne0$.

There exists a formal series $g$ such that $(1-X)g=f$, the factorial-normalized coefficients of $g$ satisfy the same rational arithmetic conditions, and $g$ is annihilated by a rational polynomial differential operator of order $m$ with nonzero leading coefficient. Thus division by $1-X$ at the zero $1$ preserves the rational-coefficient E-function conditions.

The transformed coefficients are $q_k=(1-X)p_k-(k+1)p_{k+1}$ for $k<m$, and $q_m=(1-X)p_m$. The value-zero hypothesis is expressed as a convergent real series identity, which is appropriate for rational coefficients at $1$. No arithmetic regularity or value-lifting theorem is assumed.