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

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.

References in corpus (3)

• Representations of Coherent and Squeezed States in a $f$-deformed Fock Space
• On the dimensions of the oscillator algebras induced by orthogonal polynomials
• Unified $(q;\alpha ,\beta ,\gamma ;\nu )$-deformation of one-parametric q-deformed oscillator algebras