NIST Handbook Mathematical Functions: what is the binomial coefficient involving negative elements? what is the terminology for the rising factorial?

Question

It is well known that if $n>0$, then https://en.wikipedia.org/wiki/Binomial_coefficient

\begin{align}\binom{-n}{k} &= \frac{-n\cdot-(n+1)\dots-(n+k-2)\cdot-(n+k-1)}{k!}\\ &=(-1)^k\;\frac{n\cdot(n+1)\cdot(n+2)\cdots (n + k - 1)}{k!}\\ &=(-1)^k\binom{n + k - 1}{k}\\ &=(-1)^k\left(\!\!\binom{n}{k}\!\!\right)\;.\end{align}

Here are some questions:

1. the above equations holds only for $n>0$, right? If $n<0$, we should adjust accorindlgy, but this does not seem to be clearly stated in https://en.wikipedia.org/wiki/Binomial_coefficient

2. Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark. The NIST Handbook of Mathematical Functions. Cambridge Univ. Press, 2010. NIST it states in section 1.2 that

\begin{align}\binom{n}{k}&= (-1)^k\;\frac{(0-n) \cdot (1-n) \cdot (2-n) \dots (k-2-n)\cdot (k-1-n)}{k!}\\ &= (-1)^k\;\frac{(-n)_k}{k!}\\ &=(-1)^k\binom{ k -n - 1}{k}\end{align}

what does the symbol $(-n)_k$ mean? Is this a standard terminology? I was not able to find its meaning or interpretation in the above NIST guide.

3. do we need to take care for the cases $n=0$ and $k \neq 0$, or $n\neq 0$ and $k =0$, or $n=0$ and $k=0$?

Answer 1

1. The above equations hold for all $n\in \mathbb C$. For example,

$$ 4=\binom{-(-4)}3=\frac{4\times 3\times 2}{3!}=(-1)^3\frac{(-2)\times (-3)\times (-4)}{3!}=(-1)^3\binom{-4+3-1}{3}=4 $$

2. $(n)_k:=\prod_{i=0}^{k-1} (n+i)$ is the rising factorial.

3. Everything works whether or not $n=0$ or $k=0$. There is no division by zero, because $0!=1$.