Constructing Krall-Hahn orthogonal 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 $D_p(p_n)={\theta }_n{p}_n$, 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 ${\beta }_{n,j},j=1,\dots ,m,$ 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.

arXiv comment26 pages. arXiv admin note: text overlap with arXiv:1307.1326, arXiv:1407.6973