问题

勒让德多项式(Legendre Polynomial)

我们称

\[
P_0(x),\  P_1(x),\  P_2(x),\ \ldots,\ P_n(x)
\]

为勒让德多项式(Legendre Polynomial), 如果它们满足下面的条件:

(1)  $P_n(x)$ 是 $n$ 阶实系数多项式, 且满足

\[
\int_{-1}^{1}P_n(x)x^{\nu}\mathrm{d}x=0,\quad\nu=0,1,2,\ldots,n-1;\quad n\geqslant 1;
\] 

(2)  $P_n(x)$ 满足

\[
\int_{-1}^{1}\bigl[P_n(x)\bigr]^2\mathrm{d}x=\frac{2}{2n+1},\quad n=0,1,2,\ldots;
\]

(3)  $P_n(x)$ 中 $x^n$ 的系数是正的, $n=0,1,2,\ldots$.

参考 [1], P.85

References:

[1] George Polya, Gabor Szego,  Problems and Theorems in Analysis  II.