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

Antonio J. Durán

paper

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

arXiv:1302.0881

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 $p_{n}$ : $q_{n} = p_{n} + β_{n} p_{n - 1}$ , $β_n\in \RR$. Using the concept of $\D$-operator, we determine the structure of the sequence $(β_{n})_{n}$ in order that the polynomials $(q_{n})_{n}$ are common eigenfunctions of a higher order difference operator. In addition, we generate sequences $(β_{n})_{n}$ 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.

43 pages