expansion in inverse factorials

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1: 6.10 Other Series Expansions

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§6.10(i) Inverse Factorial Series

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2: 6.18 Methods of Computation

… ►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. …

3: 2.9 Difference Equations

… ►For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). …

4: 10.41 Asymptotic Expansions for Large Order

… ►For expansions in inverse factorial series see Dunster et al. (1993). …

5: 24.19 Methods of Computation

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Buhler et al. (1992) uses the expansion

24.19.3 $$\frac{t^2}{\cosh t-1}=-2\sum _{n=0}^{\mathrm{\infty }}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},$$

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Symbols:

$B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $n$: integer and $t$: real or complex

Referenced by:

§24.19(ii)

Permalink:

http://dlmf.nist.gov/24.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

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6: 2.8 Differential Equations with a Parameter

… ►The form of the asymptotic expansion depends on the nature of the transition points in $𝐃$, that is, points at which $f(z)$ has a zero or singularity. … ►For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004). … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13. … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … ►However, in all cases with $\lambda >-2$ and $\lambda \ne 0$ or $\pm 1$, only uniform asymptotic approximations are available, not uniform asymptotic expansions. …

7: Bibliography O

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Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§1.8(v), §22.11, §4.27.

Encodings:

BibTeX

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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

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Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.

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External Links:

ISSN 0013-0915, Document, MathReview (Niro Yanagihara), ZentralBlatt

Referenced by:

§2.9(i).

Encodings:

BibTeX

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Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.

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External Links:

Document, MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

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Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii), §2.7(ii).

Encodings:

BibTeX

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8: 18.18 Sums

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Legendre

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Laguerre

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Hermite

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Ultraspherical

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Hermite

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9: 15.12 Asymptotic Approximations

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$\mathrm{\Re }z>\frac{1}{2}$ and ${\alpha }_{-}-\frac{1}{2}\pi +\delta \le \mathrm{ph}c\le {\alpha }_{+}+\frac{1}{2}\pi -\delta$, where

15.12.1 $${\alpha }_{\pm }=\arctan \left(\frac{\mathrm{ph}z-\mathrm{ph}\left(1-z\right)\mp \pi }{\ln |1-z^{-1}|}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable and ${\alpha }_{\pm }$

Permalink:

http://dlmf.nist.gov/15.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(ii), §15.12 and Ch.15

with $z$ restricted so that $\pm {\alpha }_{\pm }\in [0,\frac{1}{2}\pi )$.

… ►For another expansion in powers of $\lambda$ with explicit coefficients see Bissi et al. (2025, (C.4)). This reference also contains expansions for the case that both $b$ and $c$ are large. … ►For this result and an extension to an asymptotic expansion with error bounds see Jones (2001). … ►(Two errors in this reference are corrected in (15.12.9).) …

10: 22.10 Maclaurin Series

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§22.10(i) Maclaurin Series in $z$

… ►Further terms may be derived by substituting in the differential equations (22.13.13), (22.13.14), (22.13.15). The full expansions converge when . ►

§22.10(ii) Maclaurin Series in $k$ and $k^{\prime }$

… ►Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). …