Christoffel–Darboux formula

Remarks

For orthonormal polynomials the three-term recurrence reads $x p_j = \alpha_{j+1} p_{j+1} + \beta_j p_j + \alpha_j p_{j-1}$ with $\alpha_j = \frac{k_{j-1}}{k_j}$. Multiplying it at $x$ by $p_j(y)$, subtracting the same with $x$ and $y$ exchanged, and summing over $j$ makes the sum telescope to the right-hand side. The second formula is the limit $y \to x$ of the first. Example: let $n = 0$ and let $\mu$ be a probability measure with mean $m$ and variance $s^2$. Then $p_0 = 1$ and $p_1 = \frac{x - m}{s}$, and the right-hand side is $\frac{1}{1/s} \cdot \frac{(x - m)/s - (y - m)/s}{x - y} = 1$, which matches the left-hand side. The left-hand side $K_n(x, y)$ is the reproducing kernel of the polynomials of degree at most $n$ for the inner product $\int f g\, d\mu$.

Going deeper

For a student

The kernel $K_n(x, y)$ gives the orthogonal projection onto polynomials of degree at most $n$, so the formula controls partial sums of orthogonal expansions. The diagonal form shows $p_{n+1}' p_n - p_n' p_{n+1} > 0$, which implies that the zeros of $p_n$ and $p_{n+1}$ are real, simple and interlace. The numbers $\frac{1}{K_n(x, x)}$ are the Christoffel function, whose values at the zeros of $p_{n+1}$ are the Gaussian quadrature weights. In random matrix theory the kernel $K_n$ describes the correlations of eigenvalues of unitary invariant ensembles.