Lagrange formula for equally-spaced nodes

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1: 3.3 Interpolation
§3.3(i) Lagrange 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 …
2: 3.4 Differentiation
§3.4(i) Equally-SpacedNodes
The Lagrange $(n+1)$-point formula is …
Two-Point Formula
For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5). …
3: 2.2 Transcendental Equations
where $F_{0}=f_{0}$ and $sF_{s}$ ($s\geq 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of $(f(x))^{s}$ (Lagrange’s formula for the reversion of series). …
The nodes $x_{1},x_{2},\dots,x_{n}$ are prescribed, and the weights $w_{k}$ and error term $E_{n}(f)$ are found by integrating the product of the Lagrange interpolation polynomial of degree $n-1$ and $w(x)$. …
Gauss–Laguerre Formula
The nodes and weights of the 5-point complex Gauss quadrature formula (3.5.36) for $s=1$ are shown in Table 3.5.18. …
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 $k\leq 24$. …Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …
6: Bibliography G
• F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
• G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
• W. Gautschi (1983) How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.
• W. Gautschi (1992) On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.
• H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
• 7: Bibliography D
• A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
• J. J. Duistermaat (1974) Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.
• L. Durand (1978) Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.
• 8: Bibliography B
• W. Barrett (1981) 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.
• B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
• B. C. Berndt (1975b) 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.
• J. Berrut and L. N. Trefethen (2004) Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.
• R. Bo and R. Wong (1999) A uniform asymptotic formula for orthogonal polynomials associated with $\exp(-x^{4})$ . J. Approx. Theory 98, pp. 146–166.
• 9: 3.11 Approximation Techniques
If $J=n+1$, then $p_{n}(x)$ is the Lagrange interpolation polynomial for the set $x_{1},x_{2},\dots,x_{J}$3.3(i)). … Given $n+1$ distinct points $x_{k}$ in the real interval $[a,b]$, with ($a=$)$x_{0}($=b$), on each subinterval $[x_{k},x_{k+1}]$, $k=0,1,\ldots,n-1$, a low-degree polynomial is defined with coefficients determined by, for example, values $f_{k}$ and $f_{k}^{\prime}$ of a function $f$ and its derivative at the nodes $x_{k}$ and $x_{k+1}$. … For many applications a spline function is a more adaptable approximating tool than the Lagrange interpolation polynomial involving a comparable number of parameters; see §3.3(i), where a single polynomial is used for interpolating $f(x)$ on the complete interval $[a,b]$. …
10: 9.19 Approximations
• Martín et al. (1992) provides two simple formulas for approximating $\mathrm{Ai}\left(x\right)$ to graphical accuracy, one for $-\infty, the other for $0\leq x<\infty$.

• Corless et al. (1992) describe a method of approximation based on subdividing $\mathbb{C}$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, $\mathrm{Ai}'\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and $\mathrm{Ai}'\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\mathrm{Ai}\left(z\right)$, $\mathrm{Ai}'\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, $\mathrm{Bi}'\left(z\right)$.