Using [formula omitted]-operators to construct orthogonal polynomials satisfying higher order difference or differential equations

Abstract

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, βn∈R. Using the concept ofD-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.