Sharp estimates for the hypergeometric functions related to root systems of type $A$ and of rank 1

TL;DR

This paper conjectures precise estimates for Weyl-invariant hypergeometric functions related to root systems, proves the conjecture for type A and rank 1 cases, and discusses potential general validity.

Contribution

It introduces a conjecture for exact estimates of hypergeometric functions and proves it for specific root systems, advancing understanding in harmonic analysis on root systems.

Findings

Proved the conjecture for root system A_n

Confirmed the conjecture for all rank 1 cases

Presented evidence supporting the conjecture's general validity

Abstract

In this article, we conjecture exact estimates for the Weyl-invariant Opdam-Cherednik hypergeometric functions. We prove the conjecture for the root system $A_n$ and for all rank 1 cases. We provide other evidence that the conjecture might be true in general.