这是一份warwick华威大学PX277/PX281的成功案例

The corresponding polynomials $P$ are
$$
L_{k}(x)=\frac{1}{2^{k} k !} \frac{d^{k}}{d x^{k}}\left(x^{2}-1\right)^{k} \quad k=0,1,2, \ldots,
$$
which, up to a factor, are the Legendre polynomials $L_{k}$. The latter fulfil the orthorgonality relation
$$
\int_{-1}^{1} L_{i}(x) L_{j}(x) d x=\frac{2}{2 i+1} \delta_{i j},
$$
and the recursion relation
$$
(j+1) L_{j+1}(x)+j L_{j-1}(x)-(2 j+1) x L_{j}(x)=0 .
$$
It is common to choose the normalization condition
$$
L_{N}(1)=1 .
$$

PX277/PX281 COURSE NOTES ：

$$
L_{0}(x)=c_{1}
$$
with $c$ a constant. Using the normalization equation $L_{0}(1)=1$ we get that
$$
L_{0}(x)=1 .
$$
For $L_{1}(x)$ we have the general expression
$$
L_{1}(x)=a+b x
$$
and using the orthorgonality relation
$$
\int_{-1}^{1} L_{0}(x) L_{1}(x) d x=0
$$
we obtain $a=0$ and with the condition $L_{1}(1)=1$, we obtain $b=1$, yielding
$$
L_{1}(x)=x \text {. }
$$