A Class of Orthogonal Polynomials Suggested by a Trigonometric Hamiltonian: Symmetric States

Abstract

This is the second in a series of papers dealing with the sets of orthogonal polynomials generated by a trigonometric Hamiltonian. In the first of this series, a subclass of the Jacobi polynomials denoted by \({\cal T}_n^{(\mu)}(x)\) and referred to as the \({\cal T}\)-polynomial of the first kind, which arises in the investigation of the symmetric state eigenfunctions of the Hamiltonian under consideration, was examined. Another subclass of the Jacobi polynomials denoted by \({\cal U}_n^{(\mu)}(x)\) is introduced here representing the antisymmetric states, and is called in accordance the \({\cal T}\)-polynomial of the second kind. Moreover, by the derivation of the ultraspherical polynomial wavefunctions, interrelations between the \({\cal T}\)-polynomials of the first and second kinds as well as the other orthogonal polynomial systems are also emphasized.