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

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.