roots of shifted Legendre polynomial

Question

How can we find roots of shifted Legendre polynomials?

I found table of roots of Legendre polynomials but nothing about shifted ones.

Answer 1

I think the confusion is just due to the complicated definition of the Legendre-Polynomials. So let's first consider the following simple, abstract example:

If $ζ$ is a root of $f(x)$, what is the corresponding root $\tilde{ζ}$ of the shifted function $\tilde{f}(x):= f(x+42)$, with shift $τ(x)=x+42$?

It is: \begin{align*}\tilde{f}(\tilde{ζ}) = f(\tilde{ζ}+42) \overset{!}{=}f(ζ)=0 \end{align*} so $\tilde{ζ}=ζ-42= τ^{-1}(ζ)$. As you see, you don't need to know anything about $f$, except of the roots. The only important information is the shift

Therefore, the same works for the shifted Legendre-Polynomials, without knowing their exact definition. It is $$\tilde{P_n}(x)=P_n(2x-1).$$

Here the shift is $τ(x)=2x-1$ and the inverse is $τ^{-1}(y)=\frac{y+1}{2}$.