Sum of Lagrange polynomial

Question

I have to calculate $\sum_{i=0}^n x^k_i*l_i(x)$ for $k=0,1,2,...,n$ . I've proved that $\sum_{i=0}^n l_i(x) = 1$, but I cannot figure out how it may help me calculating $\sum_{i=0}^n x^k_i*l_i(x)$ . I'd be grateful if anyone could point me to the solution, thanks in advnace.

Answer 1

Let $k \in \mathbb N$

Define $\displaystyle L_k(x)=\sum_{i=0}^n x_i^k l_i(x)$

$L_k$ is a degree $n$ polynomial such that $\forall i \in \{0,\ldots,n\},L_k(x_i)=x_i^k$

Consider the polynomial $L_k-X^k$

It has degree less than $n$, and it has also $n+1$ roots.

Therefore, $L_k=X^k$