Transcendence and the Carlitz–Goss Gamma Function

AbstractIn this article, we show that a value of the Carlitz–Goss gamma function for the ringFq[X] is transcendental over the fieldFq(X) if and only if the argument is not an element of N={0, 1, 2, …}. We also answer a question of J.-P. Allouche, and we show the transcendence of monomials built on the values of the Carlitz–Goss gamma function. This concludes the transcendence study of the values of this function initiated and developed by D. Thakur and pursued by A. Thiery, J. Yu, and L. Denis and J.-P. Allouche. The proof of our result, as that of J.-P. Allouche, uses derivation of formal power series and the theorem of G. Christol, T. Kamae, M. Mendès France, and G. Rauzy