拉格朗日插值多项式之间的递推关系

Let $n$ be a positive integer,and let $f(x)$ be a function defined on a domain containing the $n+1$ distinct points $x_0,x_1,\cdots,x_n$,and let $p_n(x)$ be the polynomial of degree $n$ that interpolates $f(x)$ at the points $x_0,x_1,\cdots,x_n$.For each $i=0,1,\cdots,n$,we define $p_{n-1,i}(x)$ to be the polynomial of degree $n-1$ that interpolates $f(x)$ at the points $x_0,x_1,\cdots,x_{i-1},x_{i+1},\cdots,x_n$.If $i$ and $j$ are distinct nonnegative integers not exceeding $n$,then

\begin{equation}
p_n(x)=\frac{(x-x_j)p_{n-1,j}(x)-(x-x_i)p_{n-1,i}(x)}{x_i-x_j}
\end{equation}

Proof:This is very easy.$p_n(x)$ and $\frac{(x-x_j)p_{n-1,j}(x)-(x-x_i)p_{n-1,i}(x)}{x_i-x_j}$ are two polynomials of the degree $n$.At each point of $x_0,x_1,\cdots,x_n$ the output of the two polynomials are equal,so the two polynomials are equal.