  Given a sequence of polynomials $(p_n)_n$, an algebra of operators $\mathcal
A$ acting in the linear space of polynomials and an operator $D_p\in \mathcal
A$ with $D_p(p_n)=\theta_np_n$, where $\theta_n$ is any arbitrary eigenvalue,
we construct a new sequence of polynomials $(q_n)_n$ by considering a linear
combination of $m+1$ consecutive $p_n$:
$q_n=p_n+\sum_{j=1}^m\beta_{n,j}p_{n-j}$. Using the concept of
$\mathcal{D}$-operator, we determine the structure of the sequences
$\beta_{n,j}, j=1,\ldots,m,$ in order that the polynomials $(q_n)_n$ are
eigenfunctions of an operator in the algebra $\mathcal A$. As an application,
from the classical discrete family of Hahn polynomials we construct orthogonal
polynomials $(q_n)_n$ which are also eigenfunctions of higher-order difference
operators.
