A $3\times 3$ singular solution to the Matrix Bochner Problem with $\mathcal{D}(W)$ not of the form $\mathbb{C}[D]$

Abstract

The Matrix Bochner Problem aims to classify weight matrices whose sequences of orthogonal polynomials are eigenfunctions of a second-order differential operator. A major breakthrough in this direction was achieved in [7], where it was shown that, under certain natural conditions on the algebra $\mathcal{D}(W)$, all solutions arise from Darboux transformations of direct sums of classical scalar weights. In this paper, we study a new $3 \times 3$ Hermite-type weight matrix and determine its algebra $\mathcal{D}(W)$ as a $\mathbb{C}[D_1]$-module generated by $\{I, D_2\}$, where $D_{1}$ and $D_{2}$ are second-order differential operators. This complete description of the algebra allows us to prove that the weight does not arise from a Darboux transformation of classical scalar weights, showing that it falls outside the classification theorem of [7]. Unlike previous examples in [3,4], which also do not fit within this classification, the algebra $\mathcal{D}(W)$ of this weight matrix is not generated by a single differential operator $D$, making it a fundamentally different case. These results complement the classification theorem of the Matrix Bochner Problem by providing a new type of singular example.