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

$\sum _{j=0}^n\frac{f_j(x)f_j(y)}{h_j}=\frac{k_n}{h_n{k}_{n+1}}\frac{f_n(y)f_{n+1}(x)-f_{n+1}(y)f_n(x)}{x-y}$

where 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 $y\to x$ limit:$$\sum _{j=0}^n\frac{f_j^2(x)}{h_j}=\frac{k_n}{h_n{k}_{n+1}}[f_{n^{\prime }+1}(x)f_n(x)-f_{n^{\prime }}(x)f_{n+1}(x)].$$

Proof

Let $p_n$ be a sequence of polynomials orthonormal with respect to a probability measure $\mu$, and define$$a_n=⟨x{p}_n,p_{n+1}⟩,b_n=⟨x{p}_n,p_n⟩,n\ge 0$$(they are called the "Jacobi parameters"), then we have the three-term recurrence$$\begin{array}{l}{p}_0(x)=1,p_1(x)=\frac{x-b_0}{a_0},\\ x{p}_n(x)=a_n{p}_{n+1}(x)+b_n{p}_n(x)+a_{n-1}{p}_{n-1}(x),n\ge 1\end{array}$$

Proof: By definition, $⟨x{p}_n,p_k⟩=⟨p_n,x{p}_k⟩$, so if $k\le n-2$, then $x{p}_k$ is a linear combination of $p_0,...,p_{n-1}$, and thus $⟨x{p}_n,p_k⟩=0$. So, to construct $p_{n+1}$, it suffices to perform Gram-Schmidt process on $x{p}_n$ 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 $\frac{k_{n+1}}{k_n}x{p}_n-p_{n+1}$ is a degree $n$ polynomial, it is perpendicular to $p_{n+1}$, and so $⟨\frac{k_{n+1}}{k_n}x{p}_n,p_{n+1}⟩=⟨p_{n+1},p_{n+1}⟩=1$. Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.

Specific cases

Hermite polynomials:

$$\sum _{k=0}^n\frac{H_k(x)H_k(y)}{k!2^k}=\frac{1}{n!2^{n+1}}\frac{H_n(y)H_{n+1}(x)-H_n(x)H_{n+1}(y)}{x-y}.$$$$\sum _{k=0}^n\frac{H{e}_k(x)H{e}_k(y)}{k!}=\frac{1}{n!}\frac{H{e}_n(y)H{e}_{n+1}(x)-H{e}_n(x)H{e}_{n+1}(y)}{x-y}.$$

Associated Legendre polynomials:

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

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