This is a question on the homework for my finite fields class. The beginning of the assignment defines the following notation:
For $i\geq1 $, define the following elements of $A=\mathbb{F}_{q}[T]$ :
$[i]=T^{q^{i}}-T$,
$L_{i}=[i][i-1]\cdots[1]$,
$D_{i}=[i][i-1]^{q}\cdots[1]^{q^{i-1}}$
and put $D_{0}=L_{0}=1$.
Let $n\geq1$. Put $A(n):=\{a \in A\mid \text{deg} (a)< n \} $
and $e_{n}(x):=\prod_{a\in A(n)}\left(x+a\right)$.
I am trying to prove: $\sum\frac{1}{b}=\frac{(-1)^{n}}{L_{n}}$ where the LHS ranges over all monic $b$ in $A$ of degree equal to $n$.
Earlier on the assignment, we showed $e_{n}(x)+D_{n}=\prod(x+b)$ where $b$ is monic with degree $n$.
I thought I would be able to take the Carlitz analog of the logarithm $\log_{c}(x)$ and then take the formal derivative and evaluate it at zero. But I've realized it is not true that $\log_{c}(xy)=\log_{c}(x)+\log_{c}(y)$. I've been stuck on this problem for some time, and can't think of any other way to approach it. I would appreciate a nudge in the right direction.