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

TL;DR

This paper introduces a new family of orthogonal polynomials linked to Jacobi matrices with trace class inverses, providing explicit formulas, orthogonality measures, and a special case involving modified q-Laguerre polynomials.

Contribution

It derives explicit formulas for these polynomials and their characteristic functions, and explores the orthogonality measure, including a novel case with modified q-Laguerre polynomials.

Findings

Explicit formulas for the polynomials and characteristic function.

Identification of the orthogonality measure for the polynomial family.

Introduction and analysis of a special case involving modified q-Laguerre polynomials.

Abstract

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.