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

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 ($\le N$) polynomial solutions as seed solutions.

50 pages. Typos are corrected. To appear in PTEP

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• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Orthogonal Polynomials from Hermitian Matrices
• Dual Christoffel transformations
• Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials
• Wronskian/Casoratian Identities and their Application to Quantum Mechanical Systems

Cited by in corpus (2)

• Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes
• Another Type of Forward and Backward Shift Relations for Orthogonal Polynomials in the Askey Scheme