  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.
