Let $P_n(x)$, $n=0,1,2,...$ be an infinite set of polynomials satisfying the three conditions
$P_n(x)$ is of degree $n$
$P_n(1)=1$ ,$n=0,1,2,...$
$\int_{-1}^{1}p_{n}(x)p_{k}(x)dx=0$, $n\neq k$
Determine the first three members of this set.
Hint: Set $P_0(x)=a_1(x)$, $P_1(x)=b_1+b_2x$ and $P_2(x)=c_1+c_2x+c_3x^2$, and solve for the unknown constans
How do I use the suggestion you give me?. I've to confess that I'm a bit lost, can anyone give me a hint ?
Thanks
The first one is just substitution $P_0(x)=a_1x^0=a_1$ and $P_0(1)=1=a_1$
The second one $P_1(x)=b_1+b_2x$ and we have $P_1(1)=b_1+b_2=1$ and $\int_{-1}^{1}P_1(x)P_0(x)dx=\int_{-1}^{1}(b_1+b_2x)(1)dx=b_1x+b_2\frac{x^2}{2}\Bigl|_{-1}^{1}=2b_1=0$ and thus $b_2=1$. Therefore $P_1(x)=x$
For the third one we have $P_2(x)=c_1+c_2x+c_3x^2$. The conditions are $P_2(1)=c_1+c_2+c_3=1$ and $\int_{-1}^{1}P_2(x)P_0(x)dx=0$ and $\int_{-1}^{1}P_2(x)P_1(x)dx=0$ and we get three equations to find three unknown $c_1,c_2,c_3$.
The equations are $P_2(1)=c_1+c_2+c_3=1$ and $$\int_{-1}^{1}P_2(x)P_0(x)dx=(c_1x+c_2\frac{x^2}{2}+c_3\frac{x^3}{3})\Bigl|_{-1}^{1}=0 \Rightarrow 2c_1+\frac{2}{3}c_3=0$$ and $$\int_{-1}^{1}P_2(x)P_1(x)dx=(c_1\frac{x^2}{2}+c_2\frac{x^3}{3}+c_3\frac{x^4}{4})\Bigl|_{-1}^{1}=\frac{2}{3}c_2=0$$
Solving these three gives us $P_2(x)=\frac{3}{2}x^2-\frac{1}{2}$