Constructing Bispectral Orthogonal Polynomials from the Classical Discrete Families of Charlier, Meixner and Krawtchouk

Abstract

Given a sequence of polynomials $$(p_n)_n$$ , an algebra of operators $${\mathcal A}$$ acting in the linear space of polynomials, and an operator $$D_p\in {\mathcal A}$$ with $$D_p(p_n)=np_n$$ , we form a new sequence of polynomials $$(q_n)_n$$ by considering a linear combination of $$m+1$$ consecutive $$p_n$$ : $$q_n=p_n+\sum _{j=1}^m\beta _{n,j}p_{n-j}$$ . Using the concept of $$\mathcal {D}$$ -operator, we determine the structure of the sequences $$\beta _{n,j}, j=1,\ldots ,m,$$ so that the polynomials $$(q_n)_n$$ are eigenfunctions of an operator in the algebra $${\mathcal A}$$ . 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.