A determinant of the Artin-Hasse exponential coefficients

Abstract

We present a new determinant identity involving the coefficients of the Artin-Hasse exponential. In particular, if $E(x)=\exp ({\sum }_{k=0}^{\infty }\frac{x^{p^k}}{p^k})={\sum }_{n=0}^{\infty }{u}_n{x}^n$ is the Artin-Hasse exponential, we give, for any $\ell \ge 1$, a closed-form formula for the determinant $|u_{pi-j}{|}_{1\le i,j\le \ell }$ and show it is a $p$-adic unit.