  We find a recursive expression for the Bessel function of S. I. Gelfand for
irreducible generic representations of
$\operatorname{GL}_n\left(\mathbb{F}_q\right)$. We show that special values of
the Bessel function can be realized as the coefficients of $L$-functions
associated with exotic Kloosterman sums, and as traces of exterior powers of
Katz's exotic Kloosterman sheaves. As an application, we show that certain
polynomials, having special values of the Bessel function as their
coefficients, have all of their roots lying on the unit circle. As another
application, we show that special values of the Bessel function of the Shintani
base change of an irreducible generic representation are related to special
values of the Bessel function of the representation through Dickson
polynomials.
