Sum of Indices of Rising Factorial

Theorem

$x^{\overline {m + n} } = x^{\overline m} \paren {x + m}^{\overline n}$

where $x^{\overline m}$ denotes $x$ to the $m$ rising.

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

$=$

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

$\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 $

$=$

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

$\blacksquare$