Proof of the Rodrigues formula for the Legendre polynomials

[Happiness]

How is the below expression for $a_{n-2k}$ motivated?

I verified that the expression for $a_{n-2k}$ satisfies the recurrence relation by using $j=n-2k$ and $j+2=n-2(k-1)$ (and hence a similar expression for $a_{n-2(k-1)}$), but I don't understand how it is being motivated.

Source: http://www.phys.ufl.edu/~fry/6346/legendre.pdf

 

[jvicens]

I can see that the expression for $a_{n-2k}$ is motivated by the recurrence relation for Legendre polynomials. The recurrence relation states that $(n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x)$. By substituting $j=n-2k$ and $j+2=n-2(k-1)$, we can see that the expression for $a_{n-2k}$ satisfies the recurrence relation. This means that the expression follows the same pattern as the recurrence relation and can be used to generate the values of $a_{n-2k}$ for any given value of $k$. This is important because it allows us to calculate the values of $a_{n-2k}$ without having to go through the entire recurrence relation every time. It provides a more efficient way to calculate the values and also helps in understanding the underlying pattern and structure of the recurrence relation. Therefore, the expression for $a_{n-2k}$ is motivated by the need for a more efficient and systematic way to calculate the values of $a_{n-2k}$ while also providing insight into the recurrence relation.