I need your assistance with a corner case of this problem:
Find a recursive definition for the strings of odd length that start with "a" and end with "b" over the alphabet $\Sigma$={a,b}.
I've already tried the following, but I can't get the string "aaaab", "aaaaaab, etc.:
Base case: aab,abb $\in$ L
Recursive step: If u$\in$L, then uab,ubb $\in$L.
Your "recursive step" is not right, for a few reasons. You want to say that if $u$ is any string of odd length* then $aub\in L$, not just for strings $u\in L$. Try this grammar: $$\begin{align} S &\to aOb \\ O &\to COC \mid C \\ C &\to a \mid b \end{align}$$
$C\stackrel{*}\Longrightarrow$ the symbols of $\Sigma$, and $O\stackrel{*}\Longrightarrow$ all odd length strings in $\Sigma^*$. Finally, $S\stackrel{*}\Longrightarrow aub$ for every odd-length $u$ string in $\Sigma^*$.