Lagrange formula

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1: 2.2 Transcendental Equations

… ►where $F_0=f_0$ and $s{F}_s$ ($s\ge 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of ${(f(x))}^s$ (Lagrange’s formula for the reversion of series). …

2: 3.3 Interpolation

… ►The final expression in (3.3.1) is the Barycentric form of the Lagrange interpolation formula. … ►With an error term the Lagrange interpolation formula for $f$ is given by … ►

§3.3(iv) Newton’s Interpolation Formula

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3: 3.4 Differentiation

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§3.4(i) Equally-Spaced Nodes

►The Lagrange $(n+1)$-point formula is … ► …

4: 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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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

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How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.

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ISSN 0006-3835, Document, MathReview (G. A. Evans), ZentralBlatt

Referenced by:

§3.5(v), §9.17(iii).

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On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.

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Notes:

Orthogonal polynomials and numerical methods

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ISSN 0377-0427, Document, Link, MathReview (Mircea Ivan)

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Erratum (V1.0.28) for Table 18.3.1.

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

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External Links:

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

Referenced by:

§24.15(iii), §24.6.

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5: 27.13 Functions

… ►Lagrange (1770) proves that $g\left(2\right)=4$, and during the next 139 years the existence of $g\left(k\right)$ was shown for $k=3,4,5,6,7,8,10$. …A general formula states that … ►Explicit formulas for $r_k\left(n\right)$ have been obtained by similar methods for $k=6,8,10$, and $12$, but they are more complicated. Exact formulas for $r_k\left(n\right)$ have also been found for $k=3,5$, and $7$, and for all even $k\le 24$. …Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …

6: 3.5 Quadrature

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Gauss–Legendre Formula

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Gauss–Chebyshev Formula

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Gauss–Laguerre Formula

… ►a complex Gauss quadrature formula is available. …

7: Bibliography B

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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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External Links:

ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

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Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

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Document, MathReview (T. M. Apostol), ZentralBlatt

Referenced by:

§24.16(ii).

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Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.16(iii), §24.8(ii).

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Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.

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ISSN 0036-1445, Document, Link, MathReview (Mao Dong Ye)

Referenced by:

§3.3(i).

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A uniform asymptotic formula for orthogonal polynomials associated with $\exp (-x^4)$ . J. Approx. Theory 98, pp. 146–166.

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ISSN 0021-9045, Document, MathReview (V. L. Deshpande), ZentralBlatt

Referenced by:

§18.32.

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