 
Summary: TRANSCENDENCE OF THE CARLITZGOSS
GAMMA FUNCTION AT RATIONAL ARGUMENTS
JeanPaul Allouche
Abstract. We show that the values of the CarlitzGoss Gamma function for Fq [X]
are transcendental over Fq (X) for all arguments which are rational and not in N.
We also show the transcendence of monomials built on the values of the CarlitzGoss
Gamma function. This generalizes previous partial results due to Thakur, to Thiery,
to Yu and to Denis. Our proof uses derivation of formal power series and the theorem
of Christol, Kamae, Mend`es France and Rauzy.
Introduction
The CarlitzGoss Gamma function has been introduced by Goss to interpolate
the factorial function proposed by Carlitz for the ring of polynomials with coeffi
cients in a finite field. More precisely, (for more details see [9] and [15]), the Gamma
function is defined as follows (see [15]).
Let F q be the finite field with q elements, and let p be its characteristic. For any
integer n 2 N, let
[n] = X q n
\Gamma X; D 0 = 1; D j =
j
Y
