Orthogonal Polynomials from Hermitian Matrices

Abstract

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schrödinger equations. The hermitian matrices (factorisable Hamiltonians) are real symmetric tri-diagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalisation measures and the normalisation constants etc. are determined explicitly.

53 pages, no figures. Several sentences and a reference are added. To be published in J. Math. Phys

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Cited by in corpus (49)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
• The Universal Askey-Wilson Algebra
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
• Modification of Crum's Theorem for `Discrete' Quantum Mechanics
• Discrete Quantum Mechanics
• Multi-indexed (q-)Racah Polynomials
• Exactly Solvable Birth and Death Processes
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
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• Dual Christoffel transformations
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials
• Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism
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• A classification of sharp tridiagonal pairs
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• Exactly solvable discrete time Birth and Death processes
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