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

Abstract

Assume that $\{a_n;n\ge 0\}$ is a sequence of positive numbers and $\sum a_n^{-1}<\infty$. Let ${\alpha }_n=k{a}_n$, ${\beta }_n=a_n+k^2{a}_{n-1}$ where $k\in (0,1)$ is a parameter, and let $\{P_n(x)\}$ be an orthonormal polynomial sequence defined by the three-term recurrence \[ α_{0}P_{1}(x)+(β_{0}-x)P_{0}(x)=0,\ α_{n}P_{n+1}(x)+(β_{n}-x)P_{n}(x)+α_{n-1}P_{n-1}(x)=0 \] for $n\ge 1$, 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 $n\ge 0$. 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.