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

2016/12/28 by Makoto Kawashima, Kawashima, Makoto
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1612.08783

openalex publication_date 2016/12/28 · openalex created_date 2017/01/06 · openalex updated_date 2026/07/28

Abstract

Let p be a prime number and ℂ_p the completion of algebraic closure of ℚ_p. Let K be an algebraic number field. We fix an embedding ι_p:ℚ\hookrightarrow ℂ_p and denote K_p the completion of K with respect to the embedding ι_p. Let g(z)∈ K[[z]] and denote by M(g)(z)∈ \tfrac1zK[[\tfrac1z]] the formal Mellin transform of g(z). In this article, we prove that if M(g)(z) has a good Padé approximation, the special values M(g)(α) are convergent in K_p and irrational for infinitely many α∈ ℚ∩ (ℚ_p∖ ℤ_p) 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.

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