Abstract

Actually in the study of transcendence of formal power series arising from the Carlitz module, there are four quite powerful tools: Drinfeld modules, Wade's method, Diophantine approximation, and finite automata. An interesting question put forward by J.-P. Allouche concerns the concrete relationship between all of them. In this work, we shall show, by Wade's method, that a value of the Carlitz-Goss gamma function is transcendental if and only if the argument is not a natural number. We remark that this result was proved originally by M. Mendès France and the author with the help of automata theory. Now it remains to know whether it is possible to prove the above result respectively by Drinfeld modules and by Diophantine approximation, a question still open for the moment.