  Assume that $\{a_{n};\,n\geq0\}$ is a sequence of positive numbers and $\sum
a_{n}^{\,-1}<\infty$. 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 orthonormal
polynomial 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 an
explicit formula for $P_{n}(x)$ as well as for the characteristic function of
$J$ and describe the orthogonality measure for the polynomial sequence. As a
particular case, the modified $q$-Laguerre polynomials are introduced and
studied.
