Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields

We study the class of univariate polynomials β_k(X), introduced by Carlitz, with coefficients in the algebraic function field F_q(t) over the finite field F_q with q elements. It is implicit in the work of Carlitz that these polynomials form an F_q[t]-module basis of the ring Int(F_q[t])={f∈F_q(t)[X]|f(F_q[t])⊆F_q[t]} of integer-valued polynomials on the polynomial ring F_q[t]. This stands in close analogy to the famous fact that a Z-module basis of the ring Int(Z) is given by the binomial polynomials (Xk). We prove, for k=q^s, where s is a non-negative integer, that β_k is irreducible in Int(F_q[t]) and that it is even absolutely irreducible, that is, all of its powers β_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 β_k is not even irreducible if k is not a power of q.