Theory of Fundamental Bessel Functions of High Rank

TL;DR

This paper develops the analytic and asymptotic theory of fundamental Bessel functions for high-rank groups over real and complex fields, focusing on their integral representations and differential equations.

Contribution

It introduces a comprehensive framework for understanding fundamental Bessel functions of high rank, including asymptotic formulas and connection relations for their differential equations.

Findings

Derived asymptotic formulas for fundamental Bessel functions.

Established explicit connection formulas for Bessel differential equations.

Analyzed integral representations of high-rank Bessel functions.

Abstract

In this article, we shall study fundamental Bessel functions for $\mathrm{GL}_n(\mathbb{F})$ arising from the Vorono\"i summation formula for any rank $n$ and field $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, with focus on developing their analytic and asymptotic theory. The main implements and subjects of our study of fundamental Bessel functions are their formal integral representations and Bessel differential equations. We shall prove the asymptotic formulae for fundamental Bessel functions and explicit connection formulae for the Bessel differential equations.