On the dimensions of oscillator-like algebras induced by orthogonal polynomials: non-symmetric case

TL;DR

This paper investigates the conditions under which oscillator-like algebras derived from orthogonal polynomials have finite dimensions, focusing on non-symmetric recurrence relations and providing examples with Laguerre and Jacobi polynomials.

Contribution

It establishes necessary and sufficient conditions for the finite dimensionality of these algebras in the non-symmetric case, expanding understanding of their structure.

Findings

Finite-dimensionality characterized by conditions on recurrence coefficients

Derived conditions applicable to Laguerre and Jacobi polynomial-based algebras

Extended the theory of oscillator-like algebras to non-symmetric cases

Abstract

There is a generalized oscillator-like algebra associated with every class of orthogonal polynomials $\{\Psi_n(x)\}_{n=0}^{\infty}$, on the real line, satisfying a four term non-symmetric recurrence relation $x\Psi_n(x)=b_n\Psi_{n+1}(x)+a_n\Psi_n(x)+b_{n-1}\Psi_{n-1}(x),~\Psi_0(x)=1,~b_{-1}=0$. This note presents necessary and sufficient conditions on $a_n$ and $b_n$ for such algebras to be of finite dimension. As examples, we discuss the dimensions of oscillator-like algebras associated with Laguerre and Jacobi polynomials.