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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 commoneigenfunctions of an operator in the algebra $\mathcal{A}$. As an application,from the classical discrete families of Charlier, Meixner and Krawtchouk weconstruct orthogonal polynomials $(q_n)_n$ which are also eigenfunctions ofhigher order difference operators.

Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk