(33S) Legendre Polynomials

[Thomas Klemm]

From the aforementioned link:

\(
(n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x)
\)

Initial values are:

\(
\begin{align}
P_0(x) &= 1 \\
P_1(x) &= x \\
\end{align}
\)

I knew that we have an expert here: (49g 50g) Gegenbauer and Jacobi Polynomials

From the Wikipedia article Gegenbauer polynomials:

Quote:When α = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials.