On the dimensions of the oscillator algebras induced by orthogonal polynomials

TL;DR

This paper investigates the conditions under which oscillator algebras derived from orthogonal polynomials have finite dimensions, analyzing specific polynomial families and generalized deformed oscillator algebras.

Contribution

It provides necessary and sufficient conditions for the finite dimensionality of oscillator algebras linked to orthogonal polynomials and explores various polynomial examples and deformations.

Findings

Hermite, Legendre, and Gegenbauer oscillator algebras' dimensions analyzed

Conditions for finite-dimensional oscillator algebras established

Remarks on multi-boson system oscillator algebras included

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.