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

Abstract

Let p be an odd prime, and let ∑n=0∞anXn∈Fp[[X]] be the reduction modulo p of the Artin–Hasse exponential series. We obtain a polynomial expression for akp in terms of those arp with r<k, for even k<p3−1. A conjectural analogue covering the case of odd k<p can be stated in various polynomial forms, essentially in terms of the polynomial γ(X)=∑n=1p−2(Bn/n)Xp−n, where Bn denotes the nth Bernoulli number.We prove that γ(X) satisfies the functional equation γ(X−1)−γ(X)=£1(X)+Xp−1−wp−1 in Fp[X], where £1(X) and wp are the truncated logarithm and the Wilson quotient. This is an analogue modulo p of a functional equation, in Q[[X]], established by Zagier for the power series ∑n=1∞(Bn/n)Xn. The proof of our functional equation establishes a connection with a result of Nielsen of 1915, of which we provide a fresh proof. Our polynomial framing allows us to derive congruences for certain numerical sums involving divided Bernoulli numbers.