  In this article we introduce a new category of special functions called
fundamental Bessel functions arising from the Voronoi summation formula for
$\mathrm{GL}_n (\mathbb{R})$. The fundamental Bessel functions of rank one and
two are the oscillatory exponential functions $e^{\pm i x}$ 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.
