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Assume that $\{a_{n};\,n\geq0\}$ is a sequence of positive numbers and $\suma_{n}^{\,-1}<\infty$. Let $\alpha_{n}=ka_{n}$, $\beta_{n}=a_{n}+k^{2}a_{n-1}$where $k\in(0,1)$ is a parameter, and let $\{P_{n}(x)\}$ be an orthonormalpolynomial sequence defined by the three-term recurrence \[\alpha_{0}P_{1}(x)+(\beta_{0}-x)P_{0}(x)=0,\\alpha_{n}P_{n+1}(x)+(\beta_{n}-x)P_{n}(x)+\alpha_{n-1}P_{n-1}(x)=0 \] for$n\geq1$, with $P_{0}(x)=1$. Let $J$ be the corresponding Jacobi (tridiagonal)matrix, i.e. $J_{n,n}=\beta_{n}$, $J_{n,n+1}=J_{n+1,n}=\alpha_{n}$ for$n\geq0$. Then $J^{-1}$ exists and belongs to the trace class. We derive anexplicit formula for $P_{n}(x)$ as well as for the characteristic function of$J$ and describe the orthogonality measure for the polynomial sequence. As aparticular case, the modified $q$-Laguerre polynomials are introduced andstudied.

A family of orthogonal polynomials corresponding to Jacobi matrices with a trace class inverse