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

TL;DR

This paper classifies all 2x2 real hypergeometric Bochner pairs, linking differential operators and weight matrices, and explores their deformation and relation to classical pairs through bispectral transformations.

Contribution

It provides an explicit classification of 2x2 hypergeometric Bochner pairs and describes their moduli space and connections to classical pairs via noncommutative bispectral Darboux transformations.

Findings

Complete classification of 2x2 hypergeometric Bochner pairs

Description of a classifying space via algebraic sets

Relation to classical Bochner pairs through bispectral transformations

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.