Transcendence, automata theory and gamma functions for polynomial rings (Q2773317)

The aim of this paper is to give a characterization of the values of the \(T\)-adic Carlitz-Goss gamma function \(\Pi_T\) that are transcendental by using automata theory. The \(T\)-adic Carlitz-Goss gamma function \(\Pi_T\) was introduced by \textit{D. Goss} [see for instance, J. Reine Angew. Math. 317, 16--39 (1980; Zbl 0422.10021)] to interpolate the factorial function \(\Pi\) introduced for the ring \(\mathbb{F}_q[T]\) by \textit{L. Carlitz} [Duke Math. J. 3, 503--517 (1937; Zbl 0017.19501)]. The Carlitz-Goss gamma function takes its values in the field of Laurent formal power series with coefficients in \(\mathbb{F}_q\) (endowed with the \(T\)-adic topology), and is defined on the ring of \(p\)-adic integers \((q\) being a power of the prime \(p)\) as \(\Pi_T(n):=\prod^\infty_{j=0}(-D_{j, T})^{n_j}\), where the \(p\)-adic integer \(n\) admits as standard \(q\)-adic expansion \(n= \sum^\infty_{j=0} n_jq^j\) \((0\leq n_j\leq q-1)\).NEWLINENEWLINE The main result of the paper under review is that \(\Pi_T(n)\) is algebraic over \(\mathbb{F}_q(T)\) if and only if the \(q\)-adic expansion of \(n\) is ultimately constant. The proof is inspired by [\textit{J.-P. Allouche}, J. Number Theory 60, 318--328 (1996; Zbl 0862.11039)], who first introduced the use of automatic sequences in the study of transcendence values for Carlitz functions via Christol's theorem [\textit{G. Christol}, Theor. Comput. Sci. 9, 141--145 (1979; Zbl 0402.68044)].