We introduce the concept of \D-operators associated to a sequence of
polynomials (pn)n and an algebra \A of operators acting in the linear
space of polynomials. In this paper, we show that this concept is a powerful
tool to generate families of orthogonal polynomials which are eigenfunctions of
a higher order difference or differential operator. Indeed, given a classical
discrete family (pn)n of orthogonal polynomials (Charlier, Meixner,
Krawtchouk or Hahn), we form a new sequence of polynomials (qn)n by
considering a linear combination of two consecutive pn:
qn=pn+βnpn−1, \beta_n\in \RR. Using the concept of \D-operator,
we determine the structure of the sequence (βn)n in order that the
polynomials (qn)n are common eigenfunctions of a higher order difference
operator. In addition, we generate sequences (βn)n for which the
polynomials (qn)n are also orthogonal with respect to a measure. The same
approach is applied to the classical families of Laguerre and Jacobi
polynomials.Comment: 43 page