Product of Number by its Rising Factorial

Theorem

Let $x^{\overline n}$ denote the $n$th rising factorial power of $x$.

Then:

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

$\ds x x^{\overline n}$

$=$

$\ds \paren {x + n - n} x^{\overline n}$

$\ds $

$=$

$\ds \paren {x + n} x^{\overline n} - n x^{\overline n}$

$\ds $

$=$

$\ds \paren {x + n} \prod_{j \mathop = 0}^{n - 1} \paren {x + j} - n x^{\overline n}$

$\ds $

$=$

$\ds \prod_{j \mathop = 0}^n \paren {x + j} - n x^{\overline n}$

$\ds $

$=$

$\ds x^{\overline {n + 1} } - n x^{\overline n}$

$\blacksquare$