Need help with Rodrigues formula example in Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 3rd edition

[vgarg]

Can someone please show/explain to me the steps between the 2 circled formulas on the attached page #582 from Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 3rd edition.

Thank you!

 

[dirichlet]

We can write the recurrence relation as $K_l = \frac{2l}{2l+1} K_{l-1}$.
We get the formula on the left-hand side when we substitute $K_{l-1}$ with $K_{l-1} = \frac{2l-1}{2(l-1)+1} K_{l-1-1}$.
We can repeat this process until we get to $l = 1$ and $K_0$ (because of the assumption just below the grey box).
The part most on the right of the circle below is a compact way to write this product.

 

[vgarg]

Thank you!
Could you please explain where does the 2nd $2^l l!$ term in $2^l l! \frac{2^l l!}{(2l+1)!} 2$ in the lower circle come from? It has two $2^l l!$ terms in the numerator.

 

[dirichlet]

I don't know exactly how they arrive at that expression, but it could be due to some conversion that is related to the double factorial.
See: https://en.wikipedia.org/wiki/Double_factorial

 

[vgarg]

Thank you!
Can someone else please try to explain this to me?