Triple factorials and Airy functions

Last week I wrote in a post on multifactorials in which I said that

Double factorials come up fairly often, and sometimes triple, quadruple, or higher multifactorials do too.

This post gives a couple examples of triple factorials in practice.

One example I wrote about last week. Triple factorial comes up when evaluating the gamma function at an integer plus 1/3. As shown here,

$\Gamma\left(n + \frac{1}{3}\right) = \frac{(3n-2)!!!}{3^n} \Gamma\left(\frac{1}{3}\right)$

Another example involves solutions to Airy’s differential equation

One pair of independent solutions consists of the Airy functions Ai(x) and Bi(x). Another pair consists of the functions

$\begin{align*} f(x) &= 1 + \frac{1}{3!}x^3 + \frac{1\cdot 4}{6!}x^6 + \frac{1\cdot 4 \cdot 7}{9!}x^9 + \cdots \\ g(x) &= x + \frac{2}{4!}x^4 + \frac{2\cdot 5}{7!}x^7 + \frac{2\cdot 5 \cdot 8}{10!}x^{10} + \cdots \end{align*}$

given in A&S 10.4.3. Because both pairs of functions solve the same linear ODE, each is a linear combination of the other.

Notice that the numerators are triple factorials, and so the series above can be rewritten as

$\begin{align*} f(x) &= \sum_{n=0}^\infty \frac{(3n+1)!!!}{(3n+1)!} x^{3n}\\ g(x) &= \sum_{n=0}^\infty \frac{(3n+2)!!!}{(3n+2)!} x^{3n+1} \end{align*}$

The next post gives an example of quadruple factorials in action.