Christoffel–Darboux formula

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

[math]\displaystyle{ \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 f_j(x) is the jth 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 [math]\displaystyle{ y\to x }[/math] limit:[math]\displaystyle{ \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]\displaystyle{ p_n }[/math] be a sequence of polynomials orthonormal with respect to a probability measure [math]\displaystyle{ \mu }[/math], and define[math]\displaystyle{ 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[math]\displaystyle{ \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]\displaystyle{ \langle xp_n, p_k \rangle = \langle p_n, xp_k \rangle }[/math], so if [math]\displaystyle{ k \leq n-2 }[/math], then [math]\displaystyle{ xp_k }[/math] is a linear combination of [math]\displaystyle{ p_0, ..., p_{n-1} }[/math], and thus [math]\displaystyle{ \langle xp_n, p_k \rangle = 0 }[/math]. So, to construct [math]\displaystyle{ p_{n+1} }[/math], it suffices to perform Gram-Schmidt process on [math]\displaystyle{ xp_n }[/math] using [math]\displaystyle{ 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]\displaystyle{ f_n }[/math] to [math]\displaystyle{ p_n }[/math].

Since [math]\displaystyle{ \frac{k_{n+1}}{k_n}xp_n - p_{n+1} }[/math] is a degree [math]\displaystyle{ n }[/math] polynomial, it is perpendicular to [math]\displaystyle{ p_{n+1} }[/math], and so [math]\displaystyle{ \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

Hermite polynomials:

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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

• Turán's inequalities
• Sturm Chain

References

• Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, ISBN 978-0-521-62321-6 
• Christoffel, E. B. (1858), "Über die Gaußische Quadratur und eine Verallgemeinerung derselben." (in German), Journal für die Reine und Angewandte Mathematik 55: 61–82, doi:10.1515/crll.1858.55.61, ISSN 0075-4102, https://zenodo.org/record/1448880 
• Darboux, Gaston (1878), "Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série" (in French), Journal de Mathématiques Pures et Appliquées 4: 5–56, 377–416 
• Abramowitz, Milton; Stegun, Irene A. (1972), Handbook of Mathematical Functions, Dover Publications, Inc., New York, p. 785, Eq. 22.12.1 
• Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. (2010), NIST Handbook of Mathematical Functions, Cambridge University Press, p. 438, Eqs. 18.2.12 and 18.2.13, ISBN 978-0-521-19225-5, http://www.cambridge.org/9780521140638  (Hardback, ISBN:978-0-521-14063-8 Paperback)
• Simons, Frederik J.; Dahlen, F. A.; Wieczorek, Mark A. (2006), "Spatiospectral concentration on a sphere", SIAM Review 48 (1): 504–536, doi:10.1137/S0036144504445765, Bibcode: 2006SIAMR..48..504S