Rising Factorial as Factorial by Binomial Coefficient

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Theorem

Let $x \in \R$ be a real number. Let $n \in \Z_{\ge 0}$ be a positive integer.

$x^{\overline n} = n! \dbinom {x + n - 1} n$

where:

$x^{\overline n}$ denotes $x$ to the $n$ rising
$n!$ denotes the factorial of $n$
$\dbinom {x + n - 1} n$ denotes a binomial coefficient.


Proof

By definition of $x$ to the $n$ rising:

$x^{\overline n} = x \left({x + 1}\right) \cdots \left({x + n - 1}\right)$

By definition of the binomial coefficient of a real number:

$\dbinom {x + n - 1} n = \dfrac {\left({x + n - 1}\right) \left({x + n - 2}\right) \cdots x} {n!}$

Hence the result.

$\blacksquare$