Γ-functions

In this section we will introduce Γ-functions into the arithmetic of function fields. We do this by building on a basic, and still quite mysterious, construction of L. Carlitz in the A = F _r [T]-case. Recall that in Section 3.3 we introduced the Carlitz exponential $${e_C}\left( z \right) = \sum\limits_{j = 0}^\infty {{z^{{r^j}}}} /{D_j}$$ where $${D_0} = 1,{D_j} = \left[ j \right]{\left[ {j - 1} \right]^r} \cdots {\left[ 1 \right]^{{r^{j - 1}}}},$$ for j > 1, and $$\left[ j \right] = {T^{{r^j}}} - T $$. In Proposition 3.1.6 we showed that $${D_j} = {\text{ }}\mathop {\mathop {\mathop \prod \limits_{g \in A} }\limits_{g{\text{ monic}}} }\limits_{\deg {\text{ }}g = j} g.$$