Sum of Indices of Falling Factorial

Theorem

$x^{\underline {m + n} } = x^{\underline m} \paren {x - m}^{\underline n}$

where $x^{\underline m}$ denotes $x$ to the $m$ falling.

$\ds x^{\underline {m + n} }$

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds x^{\underline m} \paren {x - m}^{\underline n}$

$\blacksquare$