Christoffel-Darboux Identity

$$\sum_{k=0}^\infty {\phi_k(x)\phi_k(y)\over\gamma_k}={\phi_{m+1}(x)\phi_m(y)-\phi_m(x)\phi_{m+1}(y)\over a_m\gamma_m(x-y),}$$ (1)

where $\phi_k(x)$ are Orthogonal Polynomials with Weighting Function $W(x)$,

$$\gamma_m\equiv \int [\phi_m(x)]^2W(x)\,dx,$$ (2)

and

$$a_k\equiv{A_{k+1}\over A_k}$$ (3)

where $A_k$ is the Coefficient of $x^k$ in $\phi_k(x)$.

References

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, p. 322, 1956.