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. It states that


 * $$ \sum_{j=0}^n \frac{f_j(x) f_j(y)}{h_j} = \frac{k_n}{h_n k_{n+1}} \frac{f_n(y) f_{n+1}(x) - f_{n+1}(y) f_n(x)}{x - y}$$

where fj(x) is the jth term of a set of orthogonal polynomials of squared norm hj and leading coefficient kj.

There is also a "confluent form" of this identity by taking $$y\to x$$ limit:$$ \sum_{j=0}^n \frac{f_j^2(x)}{h_j} = \frac{k_n}{h_n k_{n+1}} \left[f_{n + 1}'(x)f_{n}(x) - f_{n}'(x) f_{n + 1}(x)\right].$$

Proof
Let $$p_n$$ be a sequence of polynomials orthonormal with respect to a probability measure $$\mu$$, and define$$a_{n}=\langle x p_{n},p_{n+1}\rangle,\qquad b_{n}=\langle x p_{n},p_{n}\rangle,\qquad n\geq0$$(they are called the "Jacobi parameters"), then we have the three-term recurrence $$\begin{array}{l l}\\ \end{array}$$

Proof: By definition, $$\langle xp_n, p_k \rangle = \langle p_n, xp_k \rangle$$, so if $$k \leq n-2$$, then $$xp_k$$ is a linear combination of $$p_0, ..., p_{n-1}$$, and thus $$\langle xp_n, p_k \rangle = 0$$. So, to construct $$p_{n+1}$$, it suffices to perform Gram-Schmidt process on $$xp_n$$ using $$p_n, p_{n-1}$$, which yields the desired recurrence.

Proof of Christoffel–Darboux formula:

Since both sides are unchanged by multiplying with a constant, we can scale each $$f_n$$ to $$p_n$$.

Since $$\frac{k_{n+1}}{k_n}xp_n - p_{n+1}$$ is a degree $$n$$ polynomial, it is perpendicular to $$p_{n+1}$$, and so $$\langle \frac{k_{n+1}}{k_n}xp_n, p_{n+1}\rangle = \langle p_{n+1}, p_{n+1}\rangle = 1$$. Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.

Specific cases
Hermite polynomials:

$$\sum_{k=0}^n \frac{H_k(x) H_k(y)}{k!2^k} = \frac{1}{n!2^{n+1}}\,\frac{H_n(y) H_{n+1}(x) - H_n(x) H_{n+1}(y)}{x - y}.$$$$\sum_{k=0}^n \frac{He_k(x) He_k(y)}{k!} = \frac{1}{n!}\,\frac{He_n(y) He_{n+1}(x) - He_n(x) He_{n+1}(y)}{x - y}.$$

Associated Legendre polynomials:


 * $$\begin{align} (\mu-\mu')\sum_{l=m}^L\,(2l+1)\frac{(l-m)!}{(l+m)!}\,P_{lm}(\mu)P_{lm}(\mu')=\qquad\qquad\qquad\qquad\qquad\\\frac{(L-m+1)!}{(L+m)!}\big[P_{L+1\,m}(\mu)P_{Lm}(\mu')-P_{Lm}(\mu)P_{L+1\,m}(\mu')\big].\end{align}$$