The Artin-Hasse series and Laguerre polynomials modulo a prime

For an odd prime $p$, let ${\mathrm{E}}_p(X)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{X}^n\in {\mathbb{F}}_p[[X]]$ 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 $T(X)={\sum }_{i=1}^{\mathrm{\infty }}{X}^{p^i}$ and $L_{p-1}^{(\alpha )}(X)$ denotes the (generalized) Laguerre polynomial of degree $p-1$. We prove that $G(X^p)={\sum }_{n=0}^{\mathrm{\infty }}(-1{)}^n{a}_{np}{X}^{np}$, and show that it satisfies $G(X^p)G(-X^p)T(X)=X^p\mathrm{.}$