Simplify $\prod_{i=0}^n \frac {c-i} c$

Question

Given some constants $c,n \in \mathbb Z$ I'd like to find a way to simplify $\prod_\limits{i=0}^n \frac {c-i} c$ but I can't find one.

May there is none existing... Or do you have an idea?

Answer 1

Notice that

$$\prod_{i=0}^n\frac{c-i}c=\prod_{i=0}^n\left(1-\frac ic\right)=\prod_{i=0}^n(1-ik)$$

where $k=\frac1c$.

As per this post, you will find that

$$\prod_{i=0}^n\frac{c-i}c=(-k)^n\frac{\Gamma(n+1-c)}{\Gamma(1-c)}$$

or any other form you so choose from the given answers there.

Answer 2

Consider this (c positive): $$ \prod_{i=0}^n \dfrac{c-i}{c} = \frac{1}{c^n}\prod_{i=0}^n c-i = \dfrac{c!}{c^n(c-n-1)!} $$