Christoffel Formula

Let {p_n(x)} be orthogonal polynomials associated with the distribution dalpha(x) on the interval [a,b]. Also let


(for c!=0) be a polynomial of order l which is nonnegative in this interval. Then the orthogonal polynomials {q(x)} associated with the distribution rho(x)dalpha(x) can be represented in terms of the polynomials p_n(x) as

 rho(x)q_n(x)=|p_n(x) p_(n+1)(x) ... p_(n+l)(x); p_n(x_1) p_(n+1)(x_1) ... p_(n+l)(x_1); | | ... |; p_n(x_l) p_(n+1)(x_l) ... p_(n+l)(x_l)|.

In the case of a zero x_k of multiplicity m>1, we replace the corresponding rows by the derivatives of order 0, 1, 2, ..., m-1 of the polynomials p_n(x_l), ..., p_(n+l)(x_l) at x=x_k.

See also

Christoffel-Darboux Identity, Orthogonal Polynomials

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Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 29-30, 1975.

Referenced on Wolfram|Alpha

Christoffel Formula

Cite this as:

Weisstein, Eric W. "Christoffel Formula." From MathWorld--A Wolfram Web Resource.

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