Falling and rising factorials

Remarks

Example: for $x = 4$ and $n = 2$, $4^{\underline{2}} = 4 \cdot 3 = 12$ and $4^{\overline{2}} = 4 \cdot 5 = 20$. Non-example: the product $4 \cdot 2 = 8$ is not $4^{\underline{2}}$, because its factors do not decrease by steps of $1$. For a natural number $m \ge n$, $m^{\underline{n}} = m!/(m-n)!$, and $m^{\underline{n}} = 0$ when $n > m$; see falling factorial and factorial. For a natural number $m \ge 1$, $m^{\overline{n}} = (m + n - 1)!/(m - 1)!$, and $1^{\overline{n}} = n^{\underline{n}} = n!$. The two are linked by $x^{\overline{n}} = (x + n - 1)^{\underline{n}} = (-1)^n (-x)^{\underline{n}}$. For real $x > 0$, $x^{\overline{n}} = \Gamma(x + n)/\Gamma(x)$, where $\Gamma$ is the gamma function. Other notations: $(x)_n$ is used for the falling factorial in combinatorics and for the rising factorial (the Pochhammer symbol) in special functions; the rising factorial is also written $x^{(n)}$. The q-Pochhammer symbol is a $q$-analogue of the rising factorial.