Stirling series

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1: 5.11 Asymptotic Expansions

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Terminology

►The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). …

2: Bibliography M

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A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

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3: Bibliography S

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On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.

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External Links:

ISSN 0075-4102, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§5.11(ii).

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4: 8.11 Asymptotic Approximations and Expansions

… ►This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. …

5: 4.13 Lambert $W$-Function

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4.13.10 $$W_k\left(z\right)\sim {\xi }_k-\ln {\xi }_k+\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{{\xi }_k^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln {\xi }_k\right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: integer, $m$: integer, $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1996, p. 350) and Jeffrey et al. (1995, Theorem 2).

Referenced by:

§4.13, §4.13, §4.45(iii), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E10

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Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

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4.13.11 $$W_{\pm 1}\left(x\mp 0\mathrm{i}\right)\sim -\eta -\ln \eta +\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\eta }^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln \eta \right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $m$: integer, $n$: integer, $x$: real variable and $\eta$

Notes:

See Corless et al. (1996, p. 350).

Referenced by:

§4.13, Erratum (V1.1.2) for Equation (4.13.11), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E11

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Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

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6: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

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§26.8(i) Definitions

… ► … ► $S(n,k)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\mathrm{\dots },n\}$ into exactly $k$ nonempty subsets. … ►

§26.8(v) Identities

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7: 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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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

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Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.

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Notes:

A revised and updated edition, with three new chapters, was published in 2004 by Cambridge University Press (Encyclopedia of Mathematics and its Applications, vol. 96).

External Links:

ISBN 0-521-35049-2, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

(17.9.3), Erratum (V1.0.23) for Equation (17.9.3).

Encodings:

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Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.

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External Links:

MathReview (M. S. Cheema), ZentralBlatt

Referenced by:

§26.21.

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Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:

FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

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8: Bibliography J

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

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Sur l’inversion de $y^{\alpha }{e}^y$ au moyen des nombres de Stirling associés. C. R. Acad. Sci. Paris Sér. I Math. 320 (12), pp. 1449–1452.

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External Links:

ISSN 0764-4442, MathReview (F. Beukers), ZentralBlatt

Referenced by:

(4.13.10).

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Stirling Numbers, Lambert W and the Gamma Function. In Mathematical Aspects of Computer and Information Sciences, J. Blömer, I. S. Kotsireas, T. Kutsia, and D. E. Simos (Eds.), Cham, pp. 275–279.

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External Links:

ISBN 978-3-319-72453-9, ZentralBlatt

Referenced by:

§4.13.

Encodings:

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.

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Notes:

Second edition of Jones and Sleeman (2003)

External Links:

ISBN 978-1-4200-8357-6, MathReview Entry, ZentralBlatt

Referenced by:

Profile Brian D. Sleeman.

Encodings:

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

A second edition was published in 2010.

External Links:

ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt

Referenced by:

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).

Encodings:

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9: Bibliography T

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Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.

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External Links:

ISBN 0-940600-05-6, FullText, MathReview (A. W. Davis), ZentralBlatt

Referenced by:

§35.4(i), §35.9.

Encodings:

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Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

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External Links:

ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§22.11.

Encodings:

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Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.

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External Links:

ISSN 0022-2526, FullText, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

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Fourier Series. Prentice-Hall Inc., Englewood Cliffs, N.J..

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Notes:

Translated from the Russian by Richard A. Silverman. Second English edition published by Dover Publications Inc., New York, 1976.

External Links:

MathReview (R. P. Boas, Jr.), ZentralBlatt

Referenced by:

§1.8(i), §1.8(ii).

Encodings:

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Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J..

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External Links:

MathReview (W. C. Rheinboldt), ZentralBlatt

Referenced by:

§3.8(iii).

Encodings:

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10: Bibliography C

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Some series and bounds for incomplete elliptic integrals. J. Math. and Phys. 40, pp. 125–134.

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External Links:

MathReview (L. Robin), ZentralBlatt

Referenced by:

§19.15, (19.5.1), (19.5.2), (19.5.4), (19.5.4_1), (19.5.4_2), (19.5.4_3), §19.9(ii).

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Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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External Links:

ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

Encodings:

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Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.

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Notes:

Reprinted by Dover, New York, 1950.

External Links:

ZentralBlatt

Referenced by:

§6.16(i).

Encodings:

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

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On a Tricomi series representation for the generalized exponential integral. Internat. J. Comput. Math. 31, pp. 257–262.

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Notes:

Includes Fortran code

External Links:

Document, ZentralBlatt

Referenced by:

§8.28(vi).

Encodings:

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