  In this paper we introduce a general family of Sz\'asz--Mirakjan--Durrmeyer
type operators depending on an integer parameter $j \in \mathbb{Z}$. They can
be viewed as a generalization of the Sz\'asz--Mirakjan--Durrmeyer operators
[9], Phillips operators [11] and corresponding Kantorovich modifications of
higher order.
  For $j\in {\mathbb{N}}$, these operators possess the exceptional property to
preserve constants and the monomial $x^{j}$. It turns out, that an extension of
this family covers certain well-known operators studied before, so that the
outcoming results could be unified.
  We present the complete asymptotic expansion for the sequence of these
operators. All its coefficients are given in a concise form. In order to prove
the expansions for the class of locally integrable functions of exponential
growth on the positive half-axis, we derive a localization result which is
interesting in itself.
