The Holonomic Rank of the Fisher-Bingham System of Differential Equations

TL;DR

This paper determines the holonomic rank of the Fisher-Bingham system of differential equations, showing it equals 2n+2, which advances understanding of its solution space structure.

Contribution

It establishes that the holonomic rank of the Fisher-Bingham system is exactly 2n+2, providing a precise measure of its solution complexity.

Findings

Holonomic rank of the Fisher-Bingham system is 2n+2.

The system is confirmed to be holonomic.

Provides a key parameter for analyzing solutions.

Abstract

The Fisher-Bingham system is a system of linear partial differential equations satisfied by the Fisher-Bingham integral for the $n$-dimensional sphere $S^n$. The system is given in [Nakayama et al. (2011), Theorem 2] and it is shown that it is a holonomic system [Koyama]. We show that the holonomic rank of the system is equal to $2n+2$.