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.
Categories: math-ph math.MP