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We find a recursive expression for the Bessel function of S. I. Gelfand forirreducible generic representations of$\operatorname{GL}_n\left(\mathbb{F}_q\right)$. We show that special values ofthe Bessel function can be realized as the coefficients of $L$-functionsassociated with exotic Kloosterman sums, and as traces of exterior powers ofKatz's exotic Kloosterman sheaves. As an application, we show that certainpolynomials, having special values of the Bessel function as theircoefficients, have all of their roots lying on the unit circle. As anotherapplication, we show that special values of the Bessel function of the Shintanibase change of an irreducible generic representation are related to specialvalues of the Bessel function of the representation through Dicksonpolynomials.

On values of the Bessel function for generic representations of finite general linear groups