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We introduce the concept of $\D$-operators associated to a sequence ofpolynomials $(p_n)_n$ and an algebra $\A$ of operators acting in the linearspace of polynomials. In this paper, we show that this concept is a powerfultool to generate families of orthogonal polynomials which are eigenfunctions ofa higher order difference or differential operator. Indeed, given a classicaldiscrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner,Krawtchouk or Hahn), we form a new sequence of polynomials $(q_n)_n$ byconsidering a linear combination of two consecutive $p_n$:$q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the concept of $\D$-operator,we determine the structure of the sequence $(\beta_n)_n$ in order that thepolynomials $(q_n)_n$ are common eigenfunctions of a higher order differenceoperator. In addition, we generate sequences $(\beta_n)_n$ for which thepolynomials $(q_n)_n$ are also orthogonal with respect to a measure. The sameapproach is applied to the classical families of Laguerre and Jacobipolynomials.

Using $\D$-operators to construct orthogonal polynomials satisfying higher order difference or differential equations