Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk

Antonio J. Durán; Manuel D. de la Iglesia

paper

Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk

arXiv:1307.1326

Abstract

Given a sequence of polynomials $(p_{n})_{n}$ , an algebra of operators $A$ acting in the linear space of polynomials and an operator $D_{p} \in A$ with $D_{p} (p_{n}) = n p_{n}$ , we form a new sequence of polynomials $(q_{n})_{n}$ by considering a linear combination of $m$ consecutive $p_{n}$ : $q_{n} = p_{n} + \sum_{j = 1}^{m} β_{n, j} p_{n - j}$ . Using the concept of $D$ -operator, we determine the structure of the sequences $β_{n, j}, j = 1, \dots, m,$ in order that the polynomials $(q_{n})_{n}$ are common eigenfunctions of an operator in the algebra $A$ . As an application, from the classical discrete families of Charlier, Meixner and Krawtchouk we construct orthogonal polynomials $(q_{n})_{n}$ which are also eigenfunctions of higher order difference operators.

35 pages

Cited by in corpus (1)

Exceptional Meixner and Laguerre orthogonal polynomials