Transcendence of gamma values for $\mathbb{F}_q[T]$


We prove that many values at proper fractions of the gamma function for $\mathbb{F}_q[T]$ (introduced by Carlitz and Goss) are transcendental over $\mathbb{F}_q(T)$. In particular, $(n-1)/b)!$ is transcendental for any integer $n$ and a positive integer $b > 1$, prime to $q$. Our proof is based on the transcendence criterion of Christol.


Dinesh S. Thakur