Fischer decompositions for entire functions of sufficiently low order

The existence of decompositions of the form $f=P\cdot q+r$ with $P_k^{\ast }\left(D\right)r=0$, where $f$ is entire, $P$ a polynomial and $P_k^{\ast }$ the principal part of $P$ with its coefficients conjugated, was achieved in \cite{AlRe23} under certain restrictions on the order of $f$. Here we prove uniqueness, thereby obtaining Fischer decompositions, under conditions that sometimes match those required for existence, and sometimes are more restrictive, depending on the parameters involved.