On infinite Jacobi matrices with a trace class resolvent

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.