Christoffel–Darboux formula

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1: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

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Confluent Form

… ►are the Christoffel numbers, see also (3.5.18). … ►

Degree lowering and raising differentiation formulas and structure relations

►For a large class of OP’s $p_n$ there exist pairs of differentiation formulas …

2: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

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Exceptional orthogonal polynomials and the Darboux transformation. J. Phys. A 43 (43), pp. 43016, 16 pp..

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ISSN 1751-8113, Document, Link, MathReview (María-José Cantero)

Referenced by:

§18.36(vi).

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

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3: 3.5 Quadrature

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§3.5(v) Gauss Quadrature

… ►In Gauss quadrature (also known as Gauss–Christoffel quadrature) we use (3.5.15) with nodes $x_k$ the zeros of $p_n$, and weights $w_k$ given by …The $w_k$ are also known as Christoffel coefficients or Christoffel numbers and they are all positive. The remainder is given by … ►

Gauss–Laguerre Formula

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4: Bibliography N

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On simplified asymptotic formulas for a class of Mathieu functions. SIAM J. Math. Anal. 15 (6), pp. 1205–1213.

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ISSN 0036-1410, Document, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§28.26(ii).

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On a simplified asymptotic formula for the Mathieu function of the third kind. SIAM J. Math. Anal. 18 (6), pp. 1616–1629.

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ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§28.26(ii).

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An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

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Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

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ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

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Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1), pp. 3–167.

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ISSN 0021-9045, Document, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.32.

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5: 14.18 Sums

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Christoffel’s Formulas

… ►In these formulas the Legendre functions are as in §14.3(ii) with $\mu =0$. The formulas are also valid with the Ferrers functions as in §14.3(i) with $\mu =0$. …

6: 2.10 Sums and Sequences

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§2.10(i) Euler–Maclaurin Formula

… ►This is the Euler–Maclaurin formula. Another version is the Abel–Plana formula: … ►

§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

… ►See also Flajolet and Odlyzko (1990).

7: Bibliography W

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Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

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Document

Referenced by:

(20.7.34), §20.7(ix).

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Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.

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FullText, Document, MathReview (S. D. Bajpai), ZentralBlatt

Referenced by:

§16.3(ii).

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The method of Darboux. J. Approximation Theory 10 (2), pp. 159–171.

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Document, MathReview (M. E. Noble), ZentralBlatt

Referenced by:

§2.10(iv).

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On a uniform treatment of Darboux’s method. Constr. Approx. 21 (2), pp. 225–255.

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External Links:

ISSN 0176-4276, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(iv).

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Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

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ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt

Referenced by:

§10.74(vii), §3.5(vii).

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8: Errata

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18.2.12 $$K_n(x,y)\equiv \sum _{\mathrm{\ell }=0}^n\frac{p_{\mathrm{\ell }}(x)p_{\mathrm{\ell }}(y)}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\frac{p_{n+1}(x)p_n(y)-p_n(x)p_{n+1}(y)}{x-y},$$ $x\ne y$

18.2.13 $$K_n(x,x)=\sum _{\mathrm{\ell }=0}^n\frac{{(p_{\mathrm{\ell }}(x))}^2}{h_{\mathrm{\ell }}}=\frac{k_n}{h_n{k}_{n+1}}\left(p_{n+1}^{\prime }(x)p_n(x)-p_n^{\prime }(x)p_{n+1}(x)\right)$$

The left-hand sides were updated to include the definition of the Christoffel–Darboux kernel $K_n(x,y)$.

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The title of the paragraph which was previously “Gasper’s $q$-Analog of Clausen’s Formula” has been changed to “Gasper’s $q$-Analog of Clausen’s Formula (16.12.2)”.

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The wording was changed to make the integration variable more apparent.

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Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

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This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

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9: 19.32 Conformal Map onto a Rectangle

… ►then $z(p)$ is a Schwartz–Christoffel mapping of the open upper-half $p$-plane onto the interior of the rectangle in the $z$-plane with vertices …

10: 31.8 Solutions via Quadratures

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$ϵ=-m_3+\frac{1}{2}$ , $m_0,m_1,m_2,m_3=0,1,2,\mathrm{\dots }$,

►the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows. …