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 Pν and Qν 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 Pν and Qν at 0 involve quadruple factorials and also Gauss’s constant.
For example, if ν = 1/2, 5/2, 9/2, …, then Pν(0) is given by
Source: An Atlas of Functions