Christoffel–Darboux formula

In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by and.
There is also a "confluent form" of this identity by taking limit:

Specific cases

Hermite

The Hermite polynomials are orthogonal with respect to the gaussian distribution.
The polynomials are orthogonal with respect to, and with.The polynomials are orthogonal with respect to, and with.The confluent form and the three-term recurrence gives

Laguerre

The Laguerre polynomials are orthonormal with respect to the exponential distribution, with, so

Legendre

Associated Legendre polynomials:

Christoffel–Darboux kernel

The summation involved in the Christoffel–Darboux formula is invariant by scaling the polynomials with nonzero constants. Thus, each probability distribution defines a series of functionswhich are called the Christoffel–Darboux kernels. By the orthogonality, the kernel satisfies In other words, the kernel is an integral operator that orthogonally projects each polynomial to the space of polynomials of degree up to.