Last week I claimed that double, triple, and quadruple factorials came up in applications. The previous post showed how triple factorials come up in solutions to Airy’s equation. This post will show how quadruple factorials come up in solutions to Legendre’s equation.

Legendre’s differential equation is

The Legendre functions *P*_{ν} and *Q*_{ν} are independent solutions to the equation for given values of ν.

When ν is of the form *n* + ½ for an integer *n*, the values of *P*_{ν} and *Q*_{ν} at 0 involve quadruple factorials and also Gauss’s constant.

For example, if ν = 1/2, 5/2, 9/2, …, then *P*_{ν}(0) is given by

Source: An Atlas of Functions