Using DD-operators to construct orthogonal polynomials satisfying higher order difference or differential equations

We introduce the concept of DD-operators associated to a sequence of polynomials (pn)n(pn)n and an algebra AA 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(pn)n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we form a new sequence of polynomials (qn)n(qn)n by considering a linear combination of two consecutive pnpn: qn=pn+βnpn−1qn=pn+βnpn−1, βn∈Rβn∈R. Using the concept ofDD-operator, we determine the structure of the sequence (βn)n(βn)n in order that the polynomials (qn)n(qn)n are common eigenfunctions of a higher order difference operator. In addition, we generate sequences (βn)n(βn)n for which the polynomials (qn)n(qn)n are also orthogonal with respect to a measure. The same approach is applied to the classical families of Laguerre and Jacobi polynomials.