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Algebraic Analysis of the Hypergeometric Function 1F1 of a Matrix Argument

Paul Görlach, Christian Lehn, Anna-Laura Sattelberger

TL;DR

The paper addresses the algebraic analysis of the hypergeometric function ${}_1F_{1}$ of a matrix argument by studying Muirhead's differential operators as a left ideal $I_m$ in the Weyl algebra and its Weyl closure $W(I_m)$. It proves that the singular locus of $I_m$ matches that of $W(I_m)$ and identifies this locus with the hyperplane arrangement ${\mathscr A}$, while conjecturing a combinatorial description of the reduced characteristic variety of $W(I_m)$ via partitions, supported by low-dimensional computations. It also discusses holonomicity, analytic solutions, and bounds for the characteristic variety, highlighting that $I_m$ is not holonomic for certain $m$ (e.g., $m=4$) but $W(I_m)$ yields holonomic systems for the matrix-argument ${}_1F_{1}$. The results provide a structured algebraic understanding of multivariate matrix-argument hypergeometric functions and suggest deeper connections to GKZ-type systems and D-module theory, with implications for both theory and computational methods in holonomic gradient frameworks.

Abstract

In this article, we investigate Muirhead's classical system of differential operators for the hypergeometric function 1F1 of a matrix argument. We formulate a conjecture for the combinatorial structure of the characteristic variety of its Weyl closure which is both supported by computational evidence as well as theoretical considerations. In particular, we determine the singular locus of this system.

Algebraic Analysis of the Hypergeometric Function 1F1 of a Matrix Argument

TL;DR

The paper addresses the algebraic analysis of the hypergeometric function of a matrix argument by studying Muirhead's differential operators as a left ideal in the Weyl algebra and its Weyl closure . It proves that the singular locus of matches that of and identifies this locus with the hyperplane arrangement , while conjecturing a combinatorial description of the reduced characteristic variety of via partitions, supported by low-dimensional computations. It also discusses holonomicity, analytic solutions, and bounds for the characteristic variety, highlighting that is not holonomic for certain (e.g., ) but yields holonomic systems for the matrix-argument . The results provide a structured algebraic understanding of multivariate matrix-argument hypergeometric functions and suggest deeper connections to GKZ-type systems and D-module theory, with implications for both theory and computational methods in holonomic gradient frameworks.

Abstract

In this article, we investigate Muirhead's classical system of differential operators for the hypergeometric function 1F1 of a matrix argument. We formulate a conjecture for the combinatorial structure of the characteristic variety of its Weyl closure which is both supported by computational evidence as well as theoretical considerations. In particular, we determine the singular locus of this system.

Paper Structure

This paper contains 18 sections, 16 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. The Weyl algebra
  3. Ideals and characteristic varieties
  4. Conormality of the characteristic variety
  5. Holonomic functions
  6. Hypergeometric functions of a matrix argument
  7. Zonal polynomials
  8. Hypergeometric functions of a matrix argument
  9. Annihilating ideals of 1F1
  10. Setup and known results about the annihilator
  11. Analytic solutions to the Muirhead ideal
  12. Characteristic variety of the Muirhead ideal
  13. Conjectural structure of the characteristic variety
  14. Bounds for the characteristic variety
  15. Examples
  16. ...and 3 more sections

Key Result

Theorem 2.3

Let $\,I\,$ be a $D_m$-ideal. Then $\,\mathop{\mathrm{Char}}\nolimits(I) \subseteq T^*{\mathbb A}^m\,$ is coisotropic. If $\,I\,$ is holonomic, every irreducible component $\,Z\,$ of the characteristic variety $\, \mathop{\mathrm{Char}}\nolimits(I) \,$ is a conormal variety. In particular, $Z$ is L

Theorems & Definitions (43)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Lemma 2.9: SST00, Theorem 1.4.15
  • ...and 33 more