$2\times 2$ Laguerre-type differential operator with triangular eigenvalue

TL;DR

This paper classifies all symmetric Laguerre-type differential operators with triangular eigenvalues related to orthogonal polynomials, introducing three families of such operators and weights with explicit parameter-dependent formulas.

Contribution

It provides a complete classification of all $2\times 2$ symmetric Laguerre-type differential operators with triangular eigenvalues, including explicit formulas for three families and a new one-parameter expression.

Findings

Three distinct families of operators and weights are characterized.

Explicit parameter-dependent formulas for each family are provided.

A new single-parameter expression for these operators is introduced.

Abstract

In this paper, we present a comprehensive account of all Laguerre-type differential operators $D$ that are symmetric with respect to a $2\times 2$ irreducible weight $W$ on the interval $(0, \infty)$. These operators are associated with monic orthogonal polynomials ${P_n}$, which satisfy the equation $DP_n = P_n\Delta_n$ for a certain lower triangular eigenvalue $\Delta_n$. We introduce three distinct families of operators and weights, each characterized by explicit expressions depending on two or three parameters, along with a new expression based on a single parameter.