Properties of Falling Factorial
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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
- $x^{\underline {m + n} } = x^{\underline m} \paren {x - m}^{\underline n}$