Factorials with arbitrary numbers enquiry?

Question

$$n(n+1)(n+2)...(n+p)$$ If I make it into a factorial, should it be $$n(n+p)!$$ or $$(n+p)!$$And can I please know the reason? Thanks

Answer 1

The most common way to notate this without the $\cdots$ is $$ \frac{(n+p)!}{(n-1)!} $$ which you should be able to convince yourself will give the intended result when $n$ is a positive integer. (Or see the comment by @cactus).

There are also shorter notations known as rising and falling factorials or Pochhammer symbols, but they are not as widely known as the ordinary notation, and some of them are ambiguous with each other, so you'd need to define explicitly which of then you're using before you do so, to avoid confusing your readers.

An advantage of the rising/falling factorial notation is that they work even when $n$ is negative or not an integer. Writing $(n+p)!/(n-1)!$ in those cases would at best be considered at best an abuse of notation, at worst nonsense.

If you just need it one or two times, it is probably not worth it to introduce rising/falling factorials explicitly, and you could simply write $$ \prod_{k=0}^p(n+k)$$