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

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 $D{P}_n=P_n\hat{I}{”}_n$ for a certain lower triangular eigenvalue $\hat{I}{”}_n$. 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.