The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function

We prove an Ax-Lindemann-Weierstrass differential transcendence result for Euler's gamma function, namely that the functions $\Gamma(\nu-\zeta_1(\nu)),\dots,\Gamma(\nu-\zeta_n(\nu))$ are differentially independent over the field of rational functions in the variable $\nu$, with coefficients in the field $k$ of $1$-periodic meromorphic functions over $\mathbb C$, as soon as $\zeta_1,\dots,\zeta_n$ determine a set of algebraic functions over $k$, stable by conjugation and pairwise distinct modulo $\mathbb Z$. \par To prove this result we use both the Galois theory of difference equations and the theory of a characteristic zero analog of the Carlitz module introduced by the second author in 2013. As an intermediate result we give an explicit description of the Picard-Vessiot rings and of the Galois groups associated to the operators in the image of the Carlitz module, using techniques inspired by the Carlitz-Hayes theory.