New finite-type multi-indexed orthogonal polynomials obtained from state-adding Darboux transformations

Abstract

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 12 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 (≤N) polynomial solutions as seed solutions.