Theory of Bessel Functions of High Rank - I: Fundamental Bessel Functions

In this article we introduce a new category of special functions called fundamental Bessel functions arising from the Voronoi summation formula for ${\mathrm{G}\mathrm{L}}_n(\mathbb{R})$. The fundamental Bessel functions of rank one and two are the oscillatory exponential functions $e^{\pm ix}$ and the classical Bessel functions respectively. The main implements and subjects of our study of fundamental Bessel functions are their formal integral representations and Bessel equations.