Can anyone tell me if there is a formula for finding the coefficient of $x^3$ in the expansion of $(3x+5)_{6}$, where $(a)_n$ denotes the Pochhammer symbol, i.e. $(a)_{n}=a\cdot(a+1)\cdots(a+n-1)$?
It would be even better if someone could explain how I might derive a general formula for the coefficient of $x^k$ in the expansion $(ax+b)_n$.
The coefficient of $x^k$ in $(x)_n$ is the signless Stirling number of the first kind, that is $c(n,k)$ the number of permutations of $n$ elements with $k$ cycles.
A lot is known about these numbers, recurrence formulas, sum formulas, generating functions, etc, and it is easy to see that this coefficient has the same recurrence formula, see http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
Now, you are looking for the coefficient of $x^k$ in $(ax+b)_n$, which gives you $$c(n,k) a^k b^{n-k}$$ because you pick up a factor $a$ for every $x$ and you pick up a factor $b$ for every parenthesis where you do not choose $x$.