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Legendre Polynomials
Definition
The Legendre polynomials are defined as
\[
\begin{align}
P_n(x) &= \frac{1}{2^n}\,\sum_{k=0}^{[n/2]} (-1)^k\,{n\choose k}\,{2n-2k\choose n}\,x^{n-2k}\\
&= {2n \choose n} \left(\frac{x}{2}\right)^n {}_2F_1 \left(\left. {-n/2+\frac{1}{2}, -n/2 \atop -n+\frac{1}{2}} \; \right| \frac{1}{x^2} \right)
\end{align}
\]
