Using $\D$-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 $(p_n)_n$ 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 $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we form a new sequence of polynomials $(q_n)_n$ by considering 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 the polynomials $(q_n)_n$ are common eigenfunctions of a higher order difference operator. In addition, we generate sequences $(\beta_n)_n$ for which the polynomials $(q_n)_n$ are also orthogonal with respect to a measure. The same approach is applied to the classical families of Laguerre and Jacobi polynomials.