Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields

We study the class of univariate polynomials ${\beta }_k(X)$, introduced by Carlitz, with coefficients in the algebraic function field ${\mathbb{F}}_q(t)$ over the finite field ${\mathbb{F}}_q$ with $q$ elements. It is implicit in the work of Carlitz that these polynomials form a ${\mathbb{F}}_q[t]$-module basis of the ring $\text{Int}({\mathbb{F}}_q[t])=\{f\in {\mathbb{F}}_q(t)[X]\mid f({\mathbb{F}}_q[t])\subseteq {\mathbb{F}}_q[t]\}$ of integer-valued polynomials on the polynomial ring ${\mathbb{F}}_q[t]$. This stands in close analogy to the famous fact that a $\mathbb{Z}$-module basis of the ring $\text{Int}(\mathbb{Z})$ is given by the binomial polynomials $\left(\genfrac{}{}{0px}{}{X}{k}\right)$. We prove, for $k=q^s$, where $s$ is a non-negative integer, that ${\beta }_k$ is irreducible in $\text{Int}({\mathbb{F}}_q[t])$ and that it is even absolutely irreducible, that is, all of its powers ${\beta }_k^m$ with $m>0$ factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that ${\beta }_k$ is not even irreducible if $k$ is not a power of $q$.