  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.
