In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel (1858) and Jean Gaston Darboux (1878). It states that
 formula_1
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 formula_2 limit:formula_3
Proof.
Let formula_4 be a sequence of polynomials orthonormal with respect to a probability measure formula_5, and defineformula_6(they are called the "Jacobi parameters"), then we have the three-term recurrenceformula_7
Proof: 
By definition, formula_8, so if formula_9, then formula_10 is a linear combination of formula_11, and thus formula_12. So, to construct formula_13, it suffices to perform Gram-Schmidt process on formula_14 using formula_15, which yields the desired recurrence.
Proof of Christoffel–Darboux formula: 
Since both sides are unchanged by multiplying with a constant, we can scale each formula_16 to formula_4.
Since formula_18 is a degree formula_19 polynomial, it is perpendicular to formula_13, and so formula_21.
Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.
Specific cases.
Hermite polynomials:
formula_22formula_23
Associated Legendre polynomials:
 formula_24
References.
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