Christoffel–Darboux formula

Template:Short description In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Template:Harvs and Template:Harvs.

There is also a "confluent form" of this identity by taking $y\to x$ limit:

Specific cases

Hermite

The Hermite polynomials are orthogonal with respect to the gaussian distribution.

The $H$ polynomials are orthogonal with respect to $\frac{1}{\sqrt{\pi }}{e}^{-x^2}$, and with $k_n=2^n$.$$\sum _{k=0}^n\frac{H_k(x)H_k(y)}{k!2^k}=\frac{1}{n!2^{n+1}}\frac{H_n(y)H_{n+1}(x)-H_n(x)H_{n+1}(y)}{x-y}.$$The $\mathrm{He}$ polynomials are orthogonal with respect to $\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{x}^2}$, and with $k_n=1$.$$\sum _{k=0}^n\frac{{\mathrm{He}}_k(x){\mathrm{He}}_k(y)}{k!}=\frac{1}{n!}\frac{{\mathrm{He}}_n(y){\mathrm{He}}_{n+1}(x)-{\mathrm{He}}_n(x){\mathrm{He}}_{n+1}(y)}{x-y}.$$The confluent form and the three-term recurrence gives$$\sum _{k=0}^n\frac{H_k(x{)}^2}{k!2^k}=\frac{H_{n+1}(x{)}^2-H_{n+2}(x)H_n(x)}{n!2^{n+1}}$$$$\sum _{k=0}^n\frac{{\mathrm{He}}_k(x{)}^2}{k!}=\frac{{\mathrm{He}}_{n+1}(x{)}^2-{\mathrm{He}}_{n+2}(x){\mathrm{He}}_n(x)}{n!}$$

Laguerre

The Laguerre polynomials $L_n$ are orthonormal with respect to the exponential distribution $e^{-x},x\in (0,\mathrm{\infty })$, with $k_n=(-1{)}^n/n!$, so$$\sum _{k=0}^n{L}_k(x)L_k(y)=\frac{n+1}{x-y}[L_n(x)L_{n+1}(y)-L_n(y)L_{n+1}(x)]$$

Legendre

Associated Legendre polynomials:

$\begin{array}{c}(\mu -{\mu }^{\prime })\sum _{l=m}^L(2l+1)\frac{(l-m)!}{(l+m)!}{P}_{lm}(\mu )P_{lm}({\mu }^{\prime })=\\ \frac{(L-m+1)!}{(L+m)!}[P_{L+1m}(\mu )P_{Lm}({\mu }^{\prime })-P_{Lm}(\mu )P_{L+1m}({\mu }^{\prime })].\end{array}$

Christoffel–Darboux kernel

The summation involved in the Christoffel–Darboux formula is invariant by scaling the polynomials with nonzero constants. Thus, each probability distribution $\mu$ defines a series of functions$$K_n(x,y):=\sum _{j=0}^n{f}_j(x)f_j(y)/h_j,n=0,1,\dots$$which are called the Christoffel–Darboux kernels. By the orthogonality, the kernel satisfies $$\int f(y)K_n(x,y)\mathrm{d}\mu (y)=\{\begin{array}{cc}f(x),& f\in \mathrm{Span}(p_0,p_1,\dots ,p_n)\\ 0,& {\int }_a^{b}f(x)p_{\ell }(x)\mathrm{d}\mu (x)=0(\ell =0,1,\dots ,n)\end{array}$$In other words, the kernel is an integral operator that orthogonally projects each polynomial to the space of polynomials of degree up to $n$.

See also

• Turán's inequalities
• Sturm's theorem

References

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