Quadruple factorials and Legendre functions

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

$(1-x^2) y'' - 2x y' + \nu(\nu+1)y = 0$

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

$(-1)^{(2\nu - 1)/4} \frac{\sqrt{2}(2\nu - 2)!!!!}{(2\nu)!!!!\pi g}$

Source: An Atlas of Functions