On the dimensions of the oscillator algebras induced by orthogonal polynomials

Abstract

There is a generalized oscillator algebra associated with every class of orthogonal polynomials $\{{\Psi }_n(x){\}}_{n=0}^{\infty }$, on the real line, satisfying a three term recurrence relation $x{\Psi }_n(x)=b_n{\Psi }_{n+1}(x)+b_{n-1}{\Psi }_{n-1}(x),{\Psi }_0(x)=1,b_{-1}=0$. This note presents necessary and safficient conditions on $b_n$ for such algebras to be of finite dimension. As examples, we discuss the dimensions of oscillator algebras associated with Hermite, Legendre and Gegenbauer polynomials. In addition we shall also discuss the dimensions of some generalized deformed oscillator algebras. Some remarks on the dimensions of oscillator algebras associated with multi-boson systems are also presented.

16 pages, J. Math. Phys (2014)

References in corpus (2)

• Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states
• Quantum deformed algebras : Coherent states and special functions

Cited by in corpus (4)

• The algebra of two dimensional generalized Chebyshev - Koornwinder oscillator
• Invariance of the generalized oscillator under linear transformation of the related system of orthogonal polynomials
• Orthogonal polynomials and deformed oscillators
• On the dimensions of oscillator-like algebras induced by orthogonal polynomials: non-symmetric case