On some coefficients of the Artin-Hasse series modulo a prime

The problem we will discuss in the talk is the calculation of the residues modulo p of the coefficients of the Artin-Hasse series without passing through their exact values as rational numbers, p an odd prime. Our results exploit a connection with another type of generalized exponentials that involves Laguerre polynomials. This leads us to prove a functional equation for the polynomial γ(X) = sum_{n=1}^{p-2} (B_n/n)X^{p-n} over the field of p elements, with Bn denoting the nth Bernoulli number. As a consequence we derive some congruences for numerical sums involving divided Bernoulli numbers.