For three consecutive orders of an orthonormal polynomial, the following relationship holds for
, 3, ...:
 |
(1)
|
where
,
,
and
are constants. Denoting the highest coefficient of
by
,
Then
 |
(5)
|
In the special case of
, (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)].](/images/equations/Christoffel-DarbouxFormula/NumberedEquation3.svg) |
(6)
|
See also
Orthonormal Functions
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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.Referenced
on Wolfram|Alpha
Christoffel-Darboux Formula
Cite this as:
Weisstein, Eric W. "Christoffel-Darboux Formula."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Christoffel-DarbouxFormula.html
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