Transcendence of gamma values for \(\mathbb{F}_ q [T]\)
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)] obtained the beautiful result that \(\Gamma (n)\) is transcendental if and only if \(n\in \mathbb{Z}_p \smallsetminus \mathbb{N}\). The last two quoted papers use again Christol's theorem and differentiation of Laurent power series.
scientific article; zbMATH DE number 935306