Transcendence and the Carlitz-Goss gamma function
Let \({\mathcal A}:=\mathbb{F}_q[T]\), \(k:=\mathbb{F}_q(T)\) and \({\mathcal K}:=\mathbb{F}_q((1/T))\). It is possible to define a ``factorial'' for \(\mathcal A\). This factorial has many of the properties of the classical \(n!\). Moreover, by taking its ``one-unit part'', this factorial interpolates to a continuous gamma function from \(\mathbb{Z}_p\) to \({\mathcal K}^*\). In the beautiful paper being reviewed, the authors establish that the values of this gamma function at all arguments NOT in \(\mathbb{N}= \{0,1,2,\dots\}\) are transcendental over \(k\). This finishes a line of research begun by D. Thakur and pursued by A. Thiery, J. Yu, L. Denis and J.-P. Allouche.
scientific article; zbMATH DE number 1011062
scientific article