# Lagrange formula for reversion of series

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##### 1: 2.2 Transcendental Equations
2.2.6 $t=y^{\frac{1}{2}}\left(1+\tfrac{1}{4}y^{-1}\ln y+o\left(y^{-1}\right)\right),$ $y\to\infty$.
An important case is the reversion of asymptotic expansions for zeros of special functions. …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). Conditions for the validity of the reversion process in $\mathbb{C}$ are derived in Olver (1997b, pp. 14–16). …
##### 2: 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 …
##### 3: 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. …
##### 4: Bibliography D
• P. Dienes (1931) The Taylor Series. Oxford University Press, Oxford.
• 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.
• T. M. Dunster (2001c) Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.
• 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.
• ##### 5: 18.40 Methods of Computation
For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. … … In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: … Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. …
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
a complex Gauss quadrature formula is available. …
##### 7: 3.4 Differentiation
###### §3.4(i) Equally-Spaced Nodes
The Lagrange $(n+1)$-point formula is …
##### 8: 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 (1968) Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.
• 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.
• ##### 9: 1.10 Functions of a Complex Variable
###### §1.10(iii) Laurent Series
The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. …
##### 10: 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.