Reference(book or article) for an explicit formula of Legendre polynomials

Question

The following explicit formula is stated for Legendre polynomials on Wikipedia.

\begin{equation}
P_n(x)=\sum_{k=0}^n {n\choose k}{n+k \choose k} \left(\dfrac{x-1}{2}\right)^2
\end{equation}

Do you know any proof or reference for this formula?

Answer 1

In Wikipedia's page there is the following Bonnet's recursion formula: $$(n+1)P_{n+1}(x)=(2n+1)xP_n(x)-nP_{n-1}(x)$$ Now, let $a_{n,k}=\binom{n}{k}\binom{n+k}{k}$. Then, esentially, we must show that $P_n(x)=\sum_{k=0}^na_{n,k}\left(\frac{x-1}{2}\right)^k$ satisfies this recursive equation. Then, we have $$(n+1)\sum_{k=0}^{n+1}a_{n+1,k}\left(\frac{x-1}{2}\right)^k=(2n+1)x\sum_{k=0}^na_{n,k}\left(\frac{x-1}{2}\right)^k-n\sum_{k=0}^{n-1}a_{n-1,k}\left(\frac{x-1}{2}\right)^k.$$ Lets do this trick: $x=2\left(\frac{x-1}{2}\right)+1$. Then we, essentially. must show that $$(n+1)a_{n+1,k}=2(2n+1)a_{n,k-1}+(2n+1)a_{n,k}-na_{n-1,k}$$ And after some simplifications, by multiplying from numerators or denominators, I got $$(n+1)(n+k+1)(n+k)=2(2n+1)k^2+(2n+1)(n-k+1)(n+k)-n(n-k+1)(n-k)$$ and these are equal.