Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields

We study the class of univariate polynomials ${\beta }_k\left(X\right)$, introduced by Carlitz, with coefficients in the algebraic function field $F_q\left(t\right)$ over the finite field $F_q$ with q elements. It is implicit in the work of Carlitz that these polynomials form an $F_q\left[t\right]$-module basis of the ring $\mathrm{Int}\left(F_q\right[t\left]\right)=\{f\in F_q(t\left)\right[X\left]\right|f\left(F_q\right[t\left]\right)\subseteq F_q\left[t\right]\}$ of integer-valued polynomials on the polynomial ring $F_q\left[t\right]$. This stands in close analogy to the famous fact that a $Z$-module basis of the ring $\mathrm{Int}\left(Z\right)$ is given by the binomial polynomials $\left(\begin{array}{c}X\\ k\end{array}\right)$.

We prove, for $k=q^s$, where s is a non-negative integer, that ${\beta }_k$ is irreducible in $\mathrm{Int}\left(F_q\right[t\left]\right)$ 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.