New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations

TL;DR

This paper introduces new finite type multi-indexed orthogonal polynomials derived from state-adding Darboux transformations in discrete quantum mechanics, expanding the class of orthogonal polynomials with explicit difference equations.

Contribution

It develops a novel method to generate multi-indexed orthogonal polynomials via higher-degree Darboux transformations, including explicit forms and difference equations for these polynomials.

Findings

New multi-indexed orthogonal polynomials satisfying second order difference equations.

Explicit forms of Krein-Adler type multi-indexed orthogonal polynomials.

Demonstration of eigenvector descriptions for deformed Hamiltonians.

Abstract

The Hamiltonians of finite type discrete quantum mechanics with real shifts are real symmetric matrices of order $N+1$. We discuss the Darboux transformations with higher degree ($>N$) polynomial solutions as seed solutions. They are state-adding and the resulting Hamiltonians after $M$-steps are of order $N+M+1$. Based on twelve orthogonal polynomials (($q$-)Racah, (dual, $q$-)Hahn, Krawtchouk and five types of $q$-Krawtchouk), new finite type multi-indexed orthogonal polynomials are obtained, which satisfy second order difference equations, and all the eigenvectors of the deformed Hamiltonian are described by them. We also present explicit forms of the Krein-Adler type multi-indexed orthogonal polynomials and their difference equations, which are obtained from the state-deleting Darboux transformations with lower degree ($\leq N$) polynomial solutions as seed solutions.