  Let $\{\hat{P}_{n}(x)\}$ be an orthonormal polynomial sequence and denote by
$\{w_{n}(x)\}$ the respective sequence of functions of the second kind. Suppose
the Hamburger moment problem for $\{\hat{P}_{n}(x)\}$ is determinate and denote
by $J$ the corresponding Jacobi matrix operator on $\ell^{2}$. We show that if
$J$ is positive definite and $J^{-1}$ belongs to the trace class then the
series on the right-hand side of the defining equation \[
\mathfrak{F}(z):=1-z\sum_{n=0}^{\infty}w_{n}(0)\hat{P}_{n}(z) \] converges
locally uniformly on $\mathbb{C}$ and it holds true that
$\mathfrak{F}(z)=\prod_{n=1}^{\infty}(1-z/\lambda_{n})$ where
$\{\lambda_{n};\,n=1,2,3,\ldots\}=\mathrm{Spec}\,J$. Furthermore, the
Al-Salam-Carlitz II polynomials are treated as an example of orthogonal
polynomials to which this theorem can be applied.
