Christoffel–Darboux formula
Introduction In mathematics, the '''Christoffel–Darboux formula''' or '''Christoffel–Darboux theorem''' is an identity for a sequence of orthogonal polynomials, introduced by  and . It states that: where ''f''''j''(''x'') is the ''j''th term of a set of orthogonal polynomials of squared norm ''h''''j'' and leading coefficient ''k''''j''.There is also a "confluent form" of this identity by taking  limit:
Proof Let  be a sequence of polynomials orthonormal with respect to a probability measure , and define(they are called the "Jacobi parameters"), then we have the three-term recurrence'''Proof:''' By definition, , so if , then  is a linear combination of , and thus . So, to construct , it suffices to perform Gram-Schmidt process on  using , which yields the desired recurrence. '''Proof of Christoffel–Darboux formula:''' Since both sides are unchanged by multiplying with a constant, we can scale each  to .Since  is a degree  polynomial, it is perpendicular to , and so .Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.
Specific cases Hermite polynomials:Associated Legendre polynomials::
See also *Turán's inequalities*Sturm Chain
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