Fischer decompositions for entire functions of sufficiently low order

The existence of decompositions of the form f=P· q+r with Pₖ^( D) r=0, where f is entire, P a polynomial and P^ₖ the principal part of P with its coefficients conjugated, was achieved in {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.