Constructing Krall-Hahn orthogonal polynomials

TL;DR

This paper develops a method to construct new orthogonal polynomials, called Krall-Hahn polynomials, that are eigenfunctions of higher-order difference operators, extending classical Hahn polynomials.

Contribution

It introduces a novel construction technique for Krall-Hahn polynomials using the concept of $\

Findings

Constructed new orthogonal polynomials from Hahn polynomials.

Established conditions for polynomials to be eigenfunctions of higher-order operators.

Extended classical Hahn polynomials to Krall-Hahn polynomials.

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 $D_p(p_n)=\theta_np_n$, where $\theta_n$ is any arbitrary eigenvalue, we construct 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 $\beta_{n,j}, j=1,\ldots,m,$ in order that the polynomials $(q_n)_n$ are eigenfunctions of an operator in the algebra $\mathcal A$. As an application, from the classical discrete family of Hahn polynomials we construct orthogonal polynomials $(q_n)_n$ which are also eigenfunctions of higher-order difference operators.