Christoffel-Darboux formula

Let ${\{{\varphi }_i\}}_{i=0}^n$ be orthonormal polynomials (the degree of ${\varphi }_k$ is $k$) and let $k_n$ be the coefficient of $x^n$ in ${\varphi }_n$. Then

$$\sum _{k=0}^n{\varphi }_k(x){\varphi }_k(y)=\frac{k_n}{k_{n+1}}\left(\frac{{\varphi }_n(y){\varphi }_{n+1}(x)-{\varphi }_n(x){\varphi }_{n+1}(y)}{x-y}\right)$$

The reason this formula is interesting is that the left-hand side isthe integral kernel for the projection operator to the subspace spannedby the polynomials ${\{{\varphi }_i\}}_{i=0}^n$.