Transcendence of the Carlitz–Goss Gamma Function at Rational Arguments Original Research Article

We show that the values of the Carlitz–Goss Gamma function for imageq[X] are transcendental over imageq(X) for all arguments which are rational and not in image. We also show the transcendence of monomials built on the values of the Carlitz–Goss Gamma function. This generalizes previous partial results due to Thakur, Thiery, Yu, and Denis. Our proof uses derivation of formal power series and the theorem of Christol, Kamae, Mendès France, and Rauzy.