Hidden double factorial example

A while back I wrote several about multi-factorials and gave examples of triple factorials and quadruple factorials in action. I didn’t do a post on double factorials because they are more common. But an example I ran into recently changed my mind.

If you skim a table of power series you will run into a lot of examples of double factorials, or rather you will run into a lot of examples that could be written in terms of double factorials. However, some sources avoid this notation. This is unfortunate because the representation in terms of double factorials is often simpler or more symmetric than the alternatives. I’ll give one example here.

If you ask Wolfram Alpha for the power series for arcsine, you’ll get several representations, the simplest being

\sin^{-1}(x) = \sum_{k=0}^\infty \frac{\left(\frac{1}{2}\right)_k}{k! + 2 k k!}x^{1 + 2k}

Here the notation (a)k denotes Pochhammer symbol, the kth rising power of a. For details, see these notes.

The series is easier to understand written in terms of double factorials:

\sin^{-1}(x) = \sum_{k=0}^\infty \frac{(2k-1)!!}{(2k)!!}\frac{x^{2k+1}}{2k+1}

Here (−1)!! is defined to be 1. This may seem odd, but the series above is an example that shows the usefulness of this definition. Also, it can be shown that

z!! = 2^{(z-1)/2} \frac{\Gamma\left(\frac{z}{2} + 1 \right)}{\Gamma\left(\frac{1}{2} + 1 \right)}

and the expression on the right equals 1 when z = −1.

However, the series above also depends on the convention that 0!! = 1. This is a common convention, and it can be justified on the grounds that it is an empty product. However, the equation above relating double factorials to the gamma function yields √(2/π) at zero.

It would be interesting to look further into why there are two candidate definitions for 0!!. It seems like an example of the relatively rare occasions where algebraic and analytic conventions conflict. I suspect the 0!! = 1 convention is usually more convenient in applications, e.g. when working with power series, though it would depend on the application.