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

Let $p$ be an odd prime, and let $\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ be the reduction modulo $p$ of the Artin-Hasse exponential. We obtain a polynomial expression for $a_{kp}$ in terms of those $a_{rp}$ with $r<k$, for even $k<p^2-1$. A conjectural analogue covering the case of odd $k<p$ can be stated in various polynomial forms, essentially in terms of the polynomial $\gamma(X) =\sum_{n=1}^{p-2}(B_{n}/n)X^{p-n}$, where $B_n$ denotes the $n$-th Bernoulli number. We prove that $\gamma(X)$ satisfies the functional equation $\gamma(X-1)-\gamma(X)=\pounds_1(X)+X^{p-1}-w_p-1$ in $\mathbb{F}_p[X]$, where $\pounds_1(X)$ and $w_p$ are the truncated logarithm and the Wilson quotient. This is an analogue modulo $p$ of a functional equation, in $\mathbb{Q}[[X]]$, established by Zagier for the power series $\sum_{n=1}^{\infty}(B_{n}/n)X^n$. Our proof of the 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.