# some formulas involving rising factorial

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

 $\Gamma(a)=\lim_{n\to\infty}\frac{n!\,n^{a-1}}{(a)_{n}}.$
Title some formulas involving rising factorial SomeFormulasInvolvingRisingFactorial 2013-03-22 17:49:12 2013-03-22 17:49:12 Wkbj79 (1863) Wkbj79 (1863) 4 Wkbj79 (1863) Result msc 05A10