Recall that, for $a\in\mathbb{C}$ and $n$ a nonnegative integer, the rising factorial $(a)_n$ is defined by $$ (a)_n=\prod_{k=0}^{n-1}(a+k). $$

The following results hold regarding the rising factorial:

• For all $a\in\mathbb{C}$ we have $(a)_0=1$
• For all nonnegative integers $n$ we have $(1)_n=n!$
• The binomial coefficients are given by $$ \binom{a}{n}=\frac{(-1)^n(-a)_n}{n!}. $$
• The rising factorial relates to the gamma function. One relation is given by the formula $$ (a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}. $$ This formula can be used to extend the definition of rising factorial so that $n$ can be any complex number provided that $a+n$ is not a nonpositive integer.
• Another relation between the rising factorial and the gamma function is given by $$ \Gamma(a)=\lim_{n\to\infty} \frac{n!\,n^{a-1}}{(a)_n}. $$