Is there an explicit polynomial form for the product of consecutive integers?

Question

I have the product $\prod_{j=0}^{r-1} (n+j)= n(n+1)\cdots(n+r-1)$ where n is a positive integer, and I was wondering if there was an explicit polynomial form for it (as a polynomial of degree r). I've tried searching for it online, and I've found expressions for when r is even and when r is odd, resulting in polynomials with coefficients that follow the central factorial number sequence, which I know nothing about. Is there a proof/paper that discusses this explicit form, this sequence, and these coefficients? Thank you.

Answer 1

This is called the "rising factorial" (also Pochhammer function, etc.)

As you've defined it, it already is a polynomial of degree $r$, and is often written $x^{(r)}$. The coefficients in terms of $n$ are not trivial to compute.

One reason these functions are interesting is that if $\Delta(f(x)) = f(x+1)-f(x),$ then $\Delta (x^{(n)}) = nx^{(n-1)},$ so the rising factorials are to $\Delta$ as $x^n$ is to $\frac{d}{dx}$.