Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial ( x ) n = x n _ = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) ⏞ n factors = ∏ k = 1 n ( x − k + 1 ) = ∏ k = 0 n − 1 ( x − k ) . {\displaystyle {\begin{aligned}(x)_{n}=x^{\underline {n}}&=\overbrace {x(x-1)(x-2)\cdots (x-n+1)} ^{n{\text{ factors}}}\\&=\prod _{k=1}^{n}(x-k+1)=\prod _{k=0}^{n-1}(x-k).\end{aligned}}} The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as x ( n ) = x n ¯ = x ( x + 1 ) ( x + 2 ) ⋯ ( x + n − 1 ) ⏞ n factors = ∏ k = 1 n ( x + k − 1 ) = ∏ k = 0 n − 1 ( x + k ) .

Source: Wikipedia — Falling and rising factorials (CC BY-SA 4.0)

Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial ( x ) n = x n _ = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) ⏞ n factors = ∏ k = 1 n ( x − k + 1 ) = ∏ k = 0 n − 1 ( x − k ) . {\displaystyle {\begin{aligned}(x)_{n}=x^{\underline {n}}&=\overbrace {x(x-1)(x-2)\cdots (x-n+1)} ^{n{\text{ factors}}}\\&=\prod _{k=1}^{n}(x-k+1)=\prod _{k=0}^{n-1}(x-k).\end{aligned}}} The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as x ( n ) = x n ¯ = x ( x + 1 ) ( x + 2 ) ⋯ ( x + n − 1 ) ⏞ n factors = ∏ k = 1 n ( x + k − 1 ) = ∏ k = 0 n − 1 ( x + k ) .

Source: Wikipedia "Falling and rising factorials" · CC BY-SA 4.0

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