Irrationality of values associated with Carlitz's exponential (Q676836)

Let \(e_C(z)\) be the exponential of the Carlitz module of \(\mathbb{F}_q[T]\) with fundamental period \(\xi\). The number \(\xi\) is analogous to \(2\pi i\) classically and one sets \(\xi_*:=\xi/(T- T^q)^{1/(q-1)}\). Let \(e:= e_C(1)\). The author has previously established that \(\{e,\xi_*\}\) are algebraically independent over \(\mathbb{F}_q(T)\) with \(q\geq 3\). In the present paper, the author sets \(q=2\) and establishes a number of results including the fact that \(\{1,e,\xi_*\}\) are linearly independent over \(\mathbb{F}_2(T)\).