Foehammer Symbol

Definitions

The Pochhammer symbol has two kinds of representations as follows.

The following equation is defined as the falling factorial.

$$ \begin{align*} x^{\underline{n}} := (x)_{n}&=x(x-1)(x-2)\cdots(x-n+1) \\ &=\frac{x!}{(x-n)!}=\frac{\Gamma (x+1) }{ \Gamma (x-n+1)} \\ &=\prod \limits_{k=0}^{n-1}(x-k) \end{align*} $$

The following equation is defined as the raising factorial.

$$ \begin{align*} x^{\overline{n}} := x^{(n)}&=x(x+1)(x+2)\cdots(x+n-1) \\ &=\frac{(x+n-1)!}{(x-1)!}=\frac{\Gamma (x+n) }{ \Gamma (x)} \\ &=\prod \limits_{k=0}^{n-1}(x+k) \end{align*} $$

$x^{\overline{0}}$ and $x^{\underline{0}}$ are defined as $1$.

$$ x^{\overline{0}}=x^{\underline{n}}=1 $$

Explanation

In combinatorial mathematics, it is a symbol that represents the product of consecutive integers. The factorial has a fixed starting number of 1. Therefore, when it is difficult or messy to express only with factorials, the Pochhammer symbol can be usefully employed. It is also used when $x$ is not an integer. There are various notations, so it is essential to check how the author has defined it in the textbook you are reading.