The Artin-Hasse series and Laguerre polynomials modulo a prime

For an odd prime $p$, let $\mathrm{E}_{p}(X)=\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ denote the reduction modulo $p$ of the Artin-Hasse exponential series. It is known that there exists a series $G(X^p)\in \mathbb{F}_{p}[[X]]$, such that $L_{p-1}^{(-T(X))}(X)=\mathrm{E}_{p}(X)\cdot G(X^p)$, where $T(X)=\sum_{i=1}^{\infty}X^{p^{i}}$ and $L_{p-1}^{(\alpha)}(X)$ denotes the (generalized) Laguerre polynomial of degree $p-1$. We prove that $G(X^p)=\sum_{n=0}^{\infty}(-1)^n a_{np}X^{np}$, and show that it satisfies $G(X^p)\,G(-X^p)\,T(X)=X^p.$