  In this contribution we consider sequences of monic polynomials orthogonal
with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle=
\langle {\bf u}^{\tt M},fg\rangle+\lambda \mathscr T^j f (\alpha)\mathscr
T^{j}g(\alpha), \] where ${\bf u}^{\tt M}$ is the Meixner linear operator,
$\lambda\in\mathbb{R}_{+}$, $j\in\mathbb{N}$, $\alpha \leq 0$, and $\mathscr T$
is the forward difference operator $\Delta$, or the backward difference
operator $\nabla$.
  We derive an explicit representation for these polynomials. The ladder
operators associated with these polynomials are obtained, and the linear
difference equation of second order is also given. In addition, for these
polynomials we derive a $(2j+3)$-term recurrence relation. Finally, we find the
Mehler-Heine type formula for the $\alpha\le 0$ case.
