Lagrange formula for reversion of series
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11: 27.20 Methods of Computation: Other Number-Theoretic Functions
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►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function for .
…To compute a particular value it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9).
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►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function , and the values can be checked by the congruence (27.14.20).
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12: 27.5 Inversion Formulas
§27.5 Inversion Formulas
►If a Dirichlet series generates , and generates , then the product generates …which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … ►Special cases of Möbius inversion pairs are: … ►Other types of Möbius inversion formulas include: …13: 30.10 Series and Integrals
14: Howard S. Cohl
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►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and -series.
Howard is the project leader for the NIST Digital Repository of Mathematical Formulae seeding and development projects.
In this regard, he has been exploring mathematical knowledge management and the digital expression of mostly unambiguous context-free full semantic information for mathematical formulae.
15: Bibliography F
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On the reversion of an asymptotic expansion and the zeros of the Airy functions.
SIAM Rev. 41 (4), pp. 762–773.
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The Edmonds asymptotic formulas for the and symbols.
J. Math. Phys. 39 (7), pp. 3906–3915.
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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series.
Chelsea Publishing Co., New York.
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On the coefficients in the recursion formulae of orthogonal polynomials.
Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.
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Series expansions of symmetric elliptic integrals.
Math. Comp. 81 (278), pp. 957–990.
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16: 3.11 Approximation Techniques
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§3.11(ii) Chebyshev-Series Expansions
… ►Summation of Chebyshev Series: Clenshaw’s Algorithm
… ►be a formal power series. … ►If , then is the Lagrange interpolation polynomial for the set (§3.3(i)). … ►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 on the complete interval . …17: 1.8 Fourier Series
§1.8 Fourier Series
… ►Parseval’s Formula
… ►Uniqueness of Fourier Series
… ►§1.8(iv) Poisson’s Summation Formula
… ►18: 29.20 Methods of Computation
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►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6.
…The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii).
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►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials.
§29.15(i) includes formulas for normalizing the eigenvectors.
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19: 36.15 Methods of Computation
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§36.15(i) Convergent Series
►Close to the origin of parameter space, the series in §36.8 can be used. … ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …20: Bibliography L
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Démonstration d’un Théoréme d’Arithmétique.
Nouveau Mém. Acad. Roy. Sci. Berlin, pp. 123–133 (French).
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Integral and series representations of the Dirac delta function.
Commun. Pure Appl. Anal. 7 (2), pp. 229–247.
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New series expansions for the confluent hypergeometric function
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Appl. Math. Comput. 235, pp. 26–31.
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New series expansions of the Gauss hypergeometric function.
Adv. Comput. Math. 39 (2), pp. 349–365.
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Expansion of the confluent hypergeometric function in series of Bessel functions.
Math. Tables Aids Comput. 13 (68), pp. 261–271.
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