The Artin-Hasse series and Laguerre polynomials modulo a prime

Abstract

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