Let $\{\phi_i\}_{i=0}^n$ be orthonormal polynomials (the degree of $\phi_k$ is $k$ ) and let $k_n$ be the coefficient of $x^n$ in $\phi_n$ . Then $$ \sum_{k=0}^n \phi_k (x) \phi_k (y) = {k_n \over k_{n+1}} \left({\phi_n (y) \phi_{n+1} (x) - \phi_n (x) \phi_{n+1} (y) \over x - y}\right $$

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