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

2013/07/04 by Durán, Antonio J., de la Iglesia, Manuel D.
#33C45 #33E30 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/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∈ 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 consecutive p_n: q_n=p_n+∑_j=1^mβ_n,jp_n-j. Using the concept of D-operator, we determine the structure of the sequences β_n,j, j=1,…,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.

Related

• Constructing Krall-Hahn orthogonal polynomials
• Using \D-operators to construct orthogonal polynomials satisfying higher order difference or differential equations
• Constructing bispectral dual Hahn polynomials
• Matrix-valued bispectral discrete orthogonal polynomials
• Automorphisms of algebras and Bochner`s property for discrete vector orthogonal polynomials