Properties of Falling Factorial

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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

$x^{\overline n} = \left({x + n - 1}\right)^{\underline n}$

Falling Factorial as Quotient of Factorials

$x^{\underline n} = \dfrac {x!} {\left({x - n}\right)!} = \dfrac {\Gamma \left({x + 1}\right)} {\Gamma \left({x - n + 1}\right)}$

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

$x^{\underline {m + n} } = x^{\underline m} \left({x - m}\right)^{\underline n}$

Also see