Factorial moment of negative binomial What's the factorial moment of negative binomial distribution, if
$$ \Pr(X = k) = \binom{k+r-1}{k} p^k(1-p)^r$$
I tried it:
$$ E\left[ \frac{X!}{(X-m)!}\right] = 
\sum_{k=m}^{\infty} \frac{(r+k-1)!}{(k-m)!} \cdot p^k = \sum_{k=0}^{\infty} \frac{(m+r-1+k)!}{k!} \cdot p^{k+m} = \sum_{k=0}^{\infty} (m+r-1)! \cdot \frac{(m+r-1+k)!}{k! \cdot (m+r-1)!} \cdot p^k \cdot p^m = p^m \cdot (m+r-1)! \cdot \sum_{k=0}^{\infty} \binom{m+r-1+k}{k} \cdot p^k. $$
What's next?
 A: Just to give an actual answer for posterity, let us use the hint by @A.S.
The probability generating function (PGF) is:
$$ f(z) = \left(\frac{1-p}{1-pz}\right)^r, $$
and thus
$$ f^{(k)}(z) = \frac{p^k \left(\frac{1-p}{1-pz}\right)^r \prod_{i=0}^{m-1} r + i}{(1-pz)^k},   $$
so finally
$$ E[(X)_m] = \frac{p^m \prod_{i=0}^{m-1} r + i}{(1-p)^m}. $$
A: While you can get the $k$th factorial moment from the generating function, you can also calculate the value directly-here's how.
First rewrite:
$$\mathbb{E}[(X)_k] = \sum_{x\geq 0} {x+r-1 \choose x} (x)_k q^xp^r,$$
by taking out $q^k$ and $p^r$ to get
$$p^rq^k\sum_{x\geq 0} {x+r-1 \choose x} (x)_k q^{x-k}.$$ Note here for brevity, I write $q=1-p$.
Next use $D^kx^a=(a)_kx^{a-k}$ where $D$ is the derivative operator.
$$p^rq^k\sum_{x\geq 0} {x+r-1 \choose x} (x)_k q^{x-k} = p^rq^k\sum_{x\geq 0} {x+r-1 \choose x} (-D)^kq^{x}.$$
Now take interchange the sum and derivative operator-you can show that this is a valid operation in this context via Fubini's theorem. Now consider,
$$p^rq^k(-D)^k \underbrace{\sum_{x\geq 0} {x+r-1 \choose x} q^{x}}_{=p^{-r}}=p^rq^k(-D)^kp^{-r}.$$
Now you can also show that $(-D)^kp^{-r} = r^{(k)}p^{-(r+k)}$, where the $r^{(k)}$ denotes a rising factorial. Using this fact gives us
$$p^rq^kr^{(k)}p^{-(r+k)},$$
which simplifies to
$$\left(\frac{q}{p}\right)^kr^{(k)}.$$
For $r=1$ this gives the formula for the $k$th factorial moment of a geometric random variable, $k!\left(\frac{q}{p}\right)^k.$
This is a-slightly-different approach than the generating function but gives you the same formula. Notice that I wrote the negative binomial as failures before the $r$th success and the OP and the other answer written give the formula where the definition is number of successes before the $r$th failure so to compare you need to change my $p$s and $q$ to match.
