Falling Factorial of Sum of Integers

Theorem

Let $r \in \R$ be a real number.

Let $a, b \in \Z$ be (positive) integers.

Then:

$r^{\underline {a + b} } = r^{\underline a} \paren {r - a}^{\underline b}$

where $r^{\underline a}$ denotes the $a$th falling factorial of $r$.

$\ds r^{\underline {a + b} }$

$=$

$\ds \prod_{j \mathop = 0}^{a + b - 1} \paren {r - j}$

$\ds $

$=$

$\ds \paren {\prod_{j \mathop = 0}^{a - 1} \paren {r - j} } \paren {\prod_{j \mathop = a}^{a + b - 1} \paren {r - j} }$

$\ds $

$=$

$\ds \paren {\prod_{j \mathop = 0}^{a - 1} \paren {r - j} } \paren {\prod_{j \mathop = 0}^{b - 1} \paren {r - a - j} }$

$\ds $

$=$

$\ds r^{\underline a} \paren {r - a}^{\underline b}$

$\blacksquare$