Irrationality of special values of formal Laurent series represented by the formal Mellin transform of $G$-functions

Abstract

Let $p$ be a prime number and $\mathbb{C}_p$ the completion of algebraic closure of $\mathbb{Q}_p$. Let $K$ be an algebraic number field. We fix an embedding $\iota_p:\overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ and denote $K_p$ the completion of $K$ with respect to the embedding $\iota_p$. Let $g(z)\in K[[z]]$ and denote by $\mathcal{M}(g)(z)\in \tfrac{1}{z}K\left[\left[\tfrac{1}{z}\right]\right]$ the formal Mellin transform of $g(z)$. In this article, we prove that if $\mathcal{M}(g)(z)$ has a good Pad\'{e} approximation, the special values $\mathcal{M}(g)(\alpha)$ are convergent in $K_p$ and irrational for infinitely many $\alpha \in \mathbb{Q}\cap \left(\mathbb{Q}_p\setminus \mathbb{Z}_p\right)$ satisfying certain conditions. This result can be regarded as a partial generalization of the method of Beukers in his proof the irrationality of special values of $p$-adic Hurwitz zeta functions.