Falling and Rising Factorials and Pochhamer Symbols

#SpecialNotations

Subjects: Special Notations, Stirling Numbers of the First Kind, Stirling Numbers of the Second Kind
We define the falling factorial is defined as

(x)_{n} = x^{\underset{―}{n}} = \prod_{k = 0}^{n - 1} (x - k) = \prod_{k = 1}^{n} (x - k + 1)

Similarly we define the rising factorial as

x^{(n)} = x^{\overset{―}{n}} = \prod_{k = 0}^{n - 1} (x + k) = \prod_{k = 1}^{n} (x + k - 1)

Properties

$(x)_{n} = (x - n + 1)^{(n)} = (- 1)^{n} (- x)^{(n)}$
$x^{(n)} = (x + n - 1)_{n} = (- 1)^{n} (- x)_{n}$

We get that the special case of the rising factorial of $1 / 2$ , we get

{(\frac{1}{2})}^{(n)} = \frac{(2 n - 1)!!}{2^{n}}

which is sometimes used in the Taylor series of $\sqrt{1 + x}$ .

We also have the following identity: $$x^{\underline k}\left(x-\frac12\right)^{\underline k} = \frac{(2x)^{\underline {2k}}}{2^{2k}}.$$

We get some really nice properties namely, with this we can define the generalised binomial coefficients as

(\binom{x}{n}) = \frac{(x)_{n}}{n!} = \frac{x^{\underset{―}{n}}}{n!}

and

(\binom{x + n - 1}{n}) = \frac{x^{(n)}}{n!} = \frac{x^{\overset{―}{n}}}{n!}

which brings more of a symmetric idea to the negative binomial coefficients.
Since

(\binom{- x}{n}) = (- 1)^{n} \frac{x^{(n)}}{n!} = (- 1)^{n} (\binom{x + n - 1}{n})

The falling factorial can be extended to real values of $x$ using the gamma function provided $x$ and $x + n$ are real numbers that are not negative integers

(x)_{n} = \frac{Γ (x + 1)}{Γ (x - n + 1)}

and the rising factorials as

x^{(n)} = \frac{Γ (x + n)}{Γ (x)}

Falling factorials appear in multiple differentiation of simple power function

{(\frac{d}{d x})}^{n} x^{a} = (a)_{n} x^{a - n}

They also appear in the hypergeomtric fucntions as

_{2} F_{1} (a, b; c; z) = \sum_{n = 0}^{\infty} \frac{a^{(n)} b^{(n)}}{c^{(n)}} \frac{z^{n}}{n!}

but the assholes use the wrong notation, when they are studying it.

Relation to the Stirling Numbers

We can make a connection to the powers using the Stirling numbers of the first kind

\begin{aligned} (x)_{n} & = \sum_{k = 0}^{n} [\begin{array}{c} n \\ k \end{array}] (- 1)^{n - k} x^{k} \\ x^{(n)} & = \sum_{k = 0}^{n} [\begin{array}{c} n \\ k \end{array}] x^{k} \end{aligned}

and similarly, using the Stirling numbers of the second kind we get that

\begin{aligned} x^{n} & = \sum_{k = 0}^{n} {\begin{array}{c} n \\ k \end{array}} (x)_{k} \\ = \sum_{k = 0}^{n} {\begin{array}{c} n \\ k \end{array}} (- 1)^{n - k} x^{(k)} . \end{aligned}

We know that the rising and falling factorials are polynomials sequences of binomial type, bringing us to a connection to Umbral calculus.

Pochhammer $k$ -symbol

We can define the Pochammer $k$ -symbol $(x)_{n, k}$ is defined as

(x)_{n, k} = \prod_{i = 0}^{n - 1} (x + i k) = k^{n} {(\frac{x}{k})}_{n}

which makes that $(x)_{n, 1} = x^{(n)}$ which is dumb, and $(x)_{n, - 1} = (x)_{n}$ which is dumber still

We can define the $k$ -gamma function as:

Γ_{k} (x) = lim_{n \to \infty} \frac{n! k^{n} (n k)^{x / k - 1}}{(x)_{n, k}}

when $k = 1$ , then $Γ_{k} = Γ$ , the usual gamma function

Properties

Relation to the Stirling Numbers

Pochhammer k-symbol

Pochhammer $k$ -symbol