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

Let $p$ be an odd prime, and let ${\sum }_{n=0}^{\infty }{a}_n{X}^n\in F_p\left[\left[X\right]\right]$ be the reduction modulo $p$ of the Artin–Hasse exponential series. We obtain a polynomial expression for $a_{kp}$ in terms of those $a_{rp}$ with $r<k$, for even $k<p^3-1$. A conjectural analogue covering the case of odd $k<p$ can be stated in various polynomial forms, essentially in terms of the polynomial $\gamma \left(X\right)={\sum }_{n=1}^{p-2}(B_n/n)X^{p-n}$, where $B_n$ denotes the $n$th Bernoulli number.

We prove that $\gamma \left(X\right)$ satisfies the functional equation $\gamma (X-1)-\gamma \left(X\right)={\pounds }_1\left(X\right)+X^{p-1}-w_p-1$ in $F_p\left[X\right]$, where ${\pounds }_1\left(X\right)$ and $w_p$ are the truncated logarithm and the Wilson quotient. This is an analogue modulo $p$ of a functional equation, in $Q\left[\left[X\right]\right]$, established by Zagier for the power series ${\sum }_{n=1}^{\infty }(B_n/n)X^n$. 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.