The Symmetric $2\times 2$ Hypergeometric Matrix Differential Operators

Abstract: We obtain an explicit classification of all $2\times 2$ real hypergeometric Bochner pairs, ie. pairs $(W(x),\mathfrak{D})$ consisting of a $2\times 2$ real hypergeometric differential operator $\mathfrak{D}$ and a $2\times 2$ weight matrix satisfying the property that $\mathfrak{D}$ is symmetric with respect to the matrix-valued inner product defined by W(x). Furthermore, we obtain a classifying space of hypergeometric Bochner pairs by describing a bijective correspondence between the collection of pairs and an open subset of a real algebraic set whose smooth paths correspond to isospectral deformations of the weight W(x) preserving a bispectral property. We also relate the hypergeometric Bochner pairs to classical Bochner pairs via noncommutative bispectral Darboux transformations.