Rising Factorial as Factorial by Binomial Coefficient

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:

$x^{\overline n} = x \paren {x + 1} \cdots \paren {x + n - 1}$

By definition of the binomial coefficient of a real number:

$\dbinom {x + n - 1} n = \dfrac {\paren {x + n - 1} \paren {x + n - 2} \cdots x} {n!}$

Hence the result.

$\blacksquare$