Properties of Falling Factorial

Theorem

Let $x^{\underline n}$ denote the $n$th falling factorial power of $x$.

This page gathers together some of the properties of the falling 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

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

Falling Factorial as Quotient of Factorials

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

Integer to Power of Itself Falling is Factorial

$n^{\underline n} = n!$

Integer to Power of Itself Less One Falling is Factorial

$n^{\underline {n - 1} } = n!$

Number to Power of One Falling is Itself

$x^{\underline 1} = x$

Number to Power of Zero Falling is One

$x^{\underline 0} = 1$

Sum of Indices of Falling Factorial

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

Also see

• Properties of Rising Factorial