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 $D_p(p_n)=np_n$, we form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of $m$ 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 common eigenfunctions of an operator in the algebra $\mathcal{ A}$. As an application, from the classical discrete families of Charlier, Meixner and Krawtchouk we construct orthogonal polynomials $(q_n)_n$ which are also eigenfunctions of higher order difference operators.
Categories: math.CA