Английская Википедия:Christoffel–Darboux formula
In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Шаблон:Harvs and Шаблон:Harvs. It states that
- <math> \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}</math>
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 <math>y\to x</math> limit:<math display="block"> \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].</math>
Proof
Let <math>p_n</math> be a sequence of polynomials orthonormal with respect to a probability measure <math>\mu</math>, and define<math display="block">a_{n}=\langle x p_{n},p_{n+1}\rangle,\qquad b_{n}=\langle x p_{n},p_{n}\rangle,\qquad n\geq0</math>(they are called the "Jacobi parameters"), then we have the three-term recurrence[1]<math display="block">\begin{array}{l l}{{p_{0}(x)=1,\qquad p_{1}(x)=\frac{x-b_{0}}{a_{0}},}}\\ {{x p_{n}(x)=a_{n}p_{n+1}(x)+b_{n}p_{n}(x)+a_{n-1}p_{n-1}(x),\qquad n\geq1}}\end{array}</math>
Proof: By definition, <math>\langle xp_n, p_k \rangle = \langle p_n, xp_k \rangle</math>, so if <math>k \leq n-2</math>, then <math>xp_k</math> is a linear combination of <math>p_0, ..., p_{n-1}</math>, and thus <math>\langle xp_n, p_k \rangle = 0</math>. So, to construct <math>p_{n+1}</math>, it suffices to perform Gram-Schmidt process on <math>xp_n</math> using <math>p_n, p_{n-1}</math>, which yields the desired recurrence.
Proof of Christoffel–Darboux formula:
Since both sides are unchanged by multiplying with a constant, we can scale each <math>f_n</math> to <math>p_n</math>.
Since <math>\frac{k_{n+1}}{k_n}xp_n - p_{n+1}</math> is a degree <math>n</math> polynomial, it is perpendicular to <math>p_{n+1}</math>, and so <math>\langle \frac{k_{n+1}}{k_n}xp_n, p_{n+1}\rangle = \langle p_{n+1}, p_{n+1}\rangle = 1</math>. Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.
Specific cases
<math display="block">\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}.</math><math display="block">\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}.</math>
Associated Legendre polynomials:
- <math>\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}</math>
See also
References
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation (Hardback, Шаблон:ISBN Paperback)
- Шаблон:Citation
