Transcendence of gamma values for \(\mathbb{F}_ q [T]\)
type: Item
instance of: scholarly article
title: Transcendence of gamma values for \(\mathbb{F}_ q [T]\)
zbMATH Open document ID: 0861.11043
DOI: 10.2307/2118588
author: Dinesh S. Thakur
published in: Annals of Mathematics. Second Series
publication date: 1997-05-12T00:00:00Z
review text: For proving the transcendence over \(\mathbb{F}_q (x)\) of a Laurent power series in \(\mathbb{F}_q ((X^{-1}))\), there are essentially four methods: Wade's method, which goes back to the 40's and has been recently developed by Damamme and Hellegouarch, the Drinfeld modules method with -- in particular -- all the results obtained by Yu, the diophantine approximation method with the works of Chérif and of de Mathan, and the finite automata method, which uses a theorem of Christol [see \textit{G. Christol}, Theor. Comput. Sci. 9, 141-145 (1979; Zbl 0402.68044) and \textit{G. Christol}, \textit{T. Kamae}, \textit{M. Mendès France} and \textit{G. Rauzy}, Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)]. An interesting class of Laurent series is given by functions, introduced by Carlitz, which are analogous, in positive characteristic, to the exponential, the logarithm, the zeta function \(\dots\) The author studies the gamma function (``Carlitz-Goss gamma function'') defined over \(\mathbb{F}_q\) by \[ \Gamma(n) = \prod_{j= 0}^\infty \left( {D_j \over X^{\deg D_j}} \right)^{n_j}, \] where \(n= \sum^\infty_0 n_jq^j\) is a \(q\)-adic integer \((0\leq n_j \leq q-1)\), and \(D_j= \prod^{j-1}_{i=0} (X^{q^j} - X^{q^i})\), \((D_0=1)\). The author proves that the values \(\Gamma (n)\) are transcendental over \(\mathbb{F}_q(X)\) for those \(n \in \mathbb{Q} \cap\mathbb{Z}_p \smallsetminus \mathbb{N}\) that satisfy an extra condition. For example \(\Gamma (n-{1 \over b})\) is transcendental for any integer \(n\) and a positive integer \(b>1\) prime to \(q\). The main tool is Christol's theorem. Note that the extra condition has been removed by the reviewer, who proved the transcendence of \(\Gamma (n)\) for \(n\in \mathbb{Q} \cap \mathbb{Z}_p \smallsetminus \mathbb{N}\), [Transcendence of the Carlitz-Goss gamma function at rational arguments, J. Number Theory 60, 318-328 (1996)]. Very recently \textit{Yao} and \textit{M. Mendès France} [J. Number Theory (to appear)] obta…
Mathematics Subject Classification ID: 11J91
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 935306
zbMATH Keywords: transcendence
zbMATH Keywords: Carlitz-Goss gamma function
zbMATH Keywords: Laurent power series
zbMATH Keywords: Christol's theorem
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118588
OpenAlex ID: W1987791080
description: scientific article; zbMATH DE number 935306