2025/03/28 by Parisi, Ignacio Bono
#33C45 #34L05 #34L10 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.22833
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 D(W), all solutions arise from Darboux transformations of direct sums of classical scalar weights. In this paper, we study a new 3 × 3 Hermite-type weight matrix and determine its algebra D(W) as a ℂ[D1]-module generated by \I, D2\, where D1 and D2 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 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.