Rising and falling powers

I’ve mentioned rising powers several times recently. They come up in practice fairly often. Sometimes you see falling powers as well, and there’s a simple symmetry between the two.

There are multiple ways to denote rising powers, and lately I’ve used the Pochhammer notation (x)_k because it’s easy to type in HTML without having to inline any LaTeX. But it’s not the best notation. It’s not mnemonic, and it’s not completely standardized.

The notation in Concrete Mathematics is mnemonic, and symmetric between rising and falling powers. This book denotes rising powers by

$x^{\overline{k}} = x(x+1)(x+2) \cdots (x + k-1)$

and falling powers by

$x^{\underline{k}} = x(x-1)(x-2) \cdots (x - k + 1)$

The 0th rising and falling power of a number is defined to be 1 because empty products are conventionally defined to be 1.

Mathematica’s notation for rising and falling powers is not at all symmetric. The kth rising power of x is denoted

    Pochhammer[x, k]

and the kth falling power is denoted

    FactorialPower[x, k].

Falling powers came up a few days ago when I wrote about the tails of a gamma distribution. That post included the asymptotic series

$P(X > x) = \frac{x^{a-1}e^{-x}}{\Gamma(a)}\left( 1 + \frac{a-1}{x} + \frac{(a-1)(a-2)}{x^2} + \cdots\right)$

The series above could be written as

$\frac{x^{a-1} e^{-x}}{\Gamma(a)} \sum_{k=0}^\infty \frac{(a-1)^{\underline{k}}}{x^k}.$

Note that we used the fact that the 0th falling power of (a-1) is 1.

You can convert between rising and falling powers by

$\begin{align*} x^{\overline{k}} &= (-1)^k (-x)^{\underline{k}} \\ x^{\underline{k}} &= (-1)^k (-x)^{\overline{k}} \end{align*}$

Some resources will define rising powers but not falling powers or vice versa because the equations above make it easy to express everything in terms of just one or the other. In my experience, probability textbooks are likely to define falling powers but not rising powers, and books on special functions are likely to define rising powers but not falling powers.