What will be the explicit formula for Shifted Legendre's polynomial in interval $x\in[a,b]$

Question

If I defined shifted Legendre polynomial $\tilde P_{n}(x)=P_{n}(\frac{2x-b-a}{b-a}) for\;all x\in[a,b]$ Then what will be the explicit formula $P_{n}(x)=\sum_{k=0}^{n} (-1)^{n+k} \frac{(n+k)!}{(n-k)! (k!)^{2}}x^{k}$ for $\tilde P_{n}(x)$?

Answer 1

QUICK ANSWER, too long for a comment, made CW.

Let $y=\frac{2x-b-a}{b-a}$, so $\tilde{P}_n(x)=P_n(y)$. You have $\tilde{P}_n(x)=P_n(y)=\sum (\text{something})y^k=\sum (\text{something})\left(\frac{2x-b-a}{b-a}\right)^k$.

Of course this is a power series in $x-\frac{b-a}{2}$.