some formulas involving rising factorial

Recall that, for $a\in ℂ$ 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 $a\in ℂ$, we have ${(a)}_0=1$.

• For all nonnegative integers $n$, we have ${(1)}_n=n!$.

• The binomial coefficients are given by

  $$\left(\genfrac{}{}{0pt}{}{a}{n}\right)=\frac{{(-1)}^n{(-a)}_n}{n!}.$$

• The rising factorial relates to the gamma function. One relation is given by the formula

  $${(a)}_n=\frac{\mathrm{\Gamma }(a+n)}{\mathrm{\Gamma }(a)}.$$

  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

  $$\mathrm{\Gamma }(a)=\underset{n\to \mathrm{\infty }}{lim}\frac{n!n^{a-1}}{{(a)}_n}.$$