Orthogonal polynomials, Legendre
The corresponding polynomials
P are
\begin{equation*}
L_k(x)=\frac{1}{2^kk!}\frac{d^k}{dx^k}(x^2-1)^k \hspace{1cm} k=0,1,2,\dots,
\end{equation*}
which, up to a factor, are the Legendre polynomials
L_k .
The latter fulfil the orthogonality relation
\begin{equation}
\int_{-1}^1L_i(x)L_j(x)dx=\frac{2}{2i+1}\delta_{ij},
\tag{10}
\end{equation}
and the recursion relation
\begin{equation}
(j+1)L_{j+1}(x)+jL_{j-1}(x)-(2j+1)xL_j(x)=0.
\tag{11}
\end{equation}