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

There is a generalized oscillator-like algebra associated with every class of orthogonal polynomials, on the real line, satisfying a four-term nonsymmetric recurrence relation. This note presents necessary and sufficient conditions, on the coefficients of the recurrence relation, for such algebras to be of finite dimension. As examples, we discuss the dimensions of oscillator-like algebras associated with Laguerre and Jacobi polynomials.