TOPICS
Search

Christoffel-Darboux Formula

The Christoffel-Darboux formula expresses a Christoffel-Darboux kernel using two consecutive orthogonal polynomials. For orthonormal polynomials with positive leading coefficients, the three-term recurrence relation holds for n=2, 3, ...:

p_n(x)=(A_nx+B_n)p_(n-1)(x)-C_np_(n-2)(x),
(1)

where A_n>0, B_n, and C_n>0 are constants. Denoting the highest coefficient of p_n(x) by k_n,

A_n=(k_n)/(k_(n-1))
(2)
C_n=(A_n)/(A_(n-1))
(3)
=(k_nk_(n-2))/(k_(n-1)^2).
(4)

Then

p_0(x)p_0(y)+...+p_n(x)p_n(y)=(k_n)/(k_(n+1))(p_(n+1)(x)p_n(y)-p_n(x)p_(n+1)(y))/(x-y).
(5)

In the special case of x=y, (5) gives

[p_0(x)]^2+...+[p_n(x)]^2=(k_n)/(k_(n+1))[p_(n+1)^'(x)p_n(x)-p_n^'(x)p_(n+1)(x)].
(6)

See also

Christoffel-Darboux Identity, Christoffel-Darboux Kernel, Orthogonal Polynomials, Orthonormal Functions

Explore with Wolfram|Alpha

References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 785, 1972.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 42-44, 1975.Trott, M. "General Properties of Orthogonal Polynomials." §2.1 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

Referenced on Wolfram|Alpha

Christoffel-Darboux Formula

Cite this as:

Weisstein, Eric W. "Christoffel-Darboux Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Christoffel-DarbouxFormula.html

Subject classifications