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Foehammer Symbol 📂Functions

Foehammer Symbol

Definitions

The Pochhammer symbol has two kinds of representations as follows.

The following equation is defined as the falling factorial.

xn‾:=(x)n=x(x−1)(x−2)⋯(x−n+1)=x!(x−n)!=Γ(x+1)Γ(x−n+1)=∏k=0n−1(x−k) \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.

xn‾:=x(n)=x(x+1)(x+2)⋯(x+n−1)=(x+n−1)!(x−1)!=Γ(x+n)Γ(x)=∏k=0n−1(x+k) \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*}

x0‾x^{\overline{0}} and x0‾x^{\underline{0}} are defined as 11.

x0‾=xn‾=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 xx 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.