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Homesome formulas involving rising factorial
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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
$(a)_{n}=\prod_{{k=0}}^{{n1}}(a+k).$ 
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
$\Gamma(a)=\lim_{{n\to\infty}}\frac{n!\,n^{{a1}}}{(a)_{n}}.$
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