A transcendency criterion in prime characteristic (Q1922476)

The author establishes a criterion for certain power series to be transcendental over \(\mathbb{F}_q(T)\), \(q\) a power of a prime \(p\). As an application the author shows the following: let \([h]: = T^{q^h}-T\in \mathbb{F}_q[T]\); let \(D_0:=1\) and for \(i>0\), \(D_i=[i]D^q_{i-1}\). The Carlitz exponential is then defined to be \(e(z): = \sum_{h>0} {z^{q^h} \over D_h}\). If one differentiates the Carlitz exponential term by term (with respect to \(T)\) \(p-1\) times one obtains \(e^{(p-1)} (z) = \sum_{h>0} {(p-1)! z^{q^h} \over [h]^{p-1} D_h}\). The author then shows that if \(0 \neq r \in \mathbb{F}_q (T)\), then \(e^{(p-1)} (r)\) is transcendental over \(\mathbb{F}_q(T)\).