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We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of GL n (F q ). 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.

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We use the Langlands-Shahidi method in order to define the Shahidi gamma factor for a pair of irreducible generic representations of GL n (F q ) and GL m (F q ). We prove that the Shahidi gamma factor is multiplicative and show that it is related to the Jacquet-Piatetski-Shapiro-Shalika gamma factor. As an application, we prove a converse theorem based on the absolute value of the Shahidi gamma factor, and improve the converse theorem of Nien. As another application, we give explicit formulas for special values of the Bessel function of an irreducible generic representation of GL n (F q ).