Pochhammer symbol

Remarks

Example: $(1)_n = 1 \cdot 2 \cdots n = n!$, the factorial; and $(2)_3 = 2 \cdot 3 \cdot 4 = 24$. For a negative integer $a = -m$ and $n > m$, one factor is $0$, so $(-m)_n = 0$; this is why hypergeometric series with such a parameter terminate. Other notations are $a^{\overline{n}}$ and $a^{(n)}$. The rising factorial of $a$ is $(-1)^n$ times the falling factorial of $-a$: $(a)_n = (-1)^n (-a)(-a-1)\cdots(-a-n+1)$. If $a \in \C$ is not in $\set{0, -1, -2, \ldots}$, then $(a)_n = \Gamma(a + n)/\Gamma(a)$, where $\Gamma$ is the gamma function. In Mathlib the rising factorial as a polynomial in $a$ is ascPochhammer.