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.