Properties of Rising Factorial

Theorem

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

This page gathers together some of the properties of the rising factorial.

Rising Factorial in terms of Falling Factorial

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

Rising Factorial in terms of Falling Factorial of Negative

$x^{\overline k} = \paren {-1}^k \paren {-x}^{\underline k}$

Rising Factorial as Quotient of Factorials

$x^{\overline n} = \dfrac {\paren {x + n - 1}!} {\paren {x - 1}!} = \dfrac {\map \Gamma {x + n} } {\map \Gamma x}$

One to Integer Rising is Integer Factorial

$1^{\overline n} = n!$

Number to Power of One Rising is Itself

$x^{\overline 1} = x$

Number to Power of Zero Rising is One

$x^{\overline 0} = 1$

Sum of Indices of Rising Factorial

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

Also see

• Properties of Falling Factorial