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Stirling series

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1: 5.11 Asymptotic Expansions
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
  • C. Mortici (2011a) A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.
  • 3: Bibliography S
  • F. W. Schäfke and A. Finsterer (1990) On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.
  • 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
    4.13.10 W k ( z ) ξ k ln ξ k + n = 1 ( 1 ) n ξ k n m = 1 n [ n n m + 1 ] ( ln ξ k ) m m ! ,
    4.13.11 W ± 1 ( x 0 i ) η ln η + n = 1 1 η n m = 1 n [ n n m + 1 ] ( ln η ) m m ! ,
    6: 26.8 Set Partitions: Stirling Numbers
    §26.8 Set Partitions: Stirling Numbers
    §26.8(i) Definitions
    S ( n , k ) denotes the Stirling number of the second kind: the number of partitions of { 1 , 2 , , n } into exactly k nonempty subsets. …
    §26.8(v) Identities
    7: Bibliography J
  • A. J. E. M. Janssen (2021) 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.
  • D. J. Jeffrey, R. M. Corless, D. E. G. Hare, and D. E. Knuth (1995) Sur l’inversion de y α e y au moyen des nombres de Stirling associés. C. R. Acad. Sci. Paris Sér. I Math. 320 (12), pp. 1449–1452.
  • D. J. Jeffrey and N. Murdoch (2017) 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.
  • D. S. Jones, M. J. Plank, and B. D. Sleeman (2010) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.
  • D. S. Jones and B. D. Sleeman (2003) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.
  • 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 and M. Rahman (1990) Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.
  • K. Goldberg, F. T. Leighton, M. Newman, and S. L. Zuckerman (1976) Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.
  • H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.
  • R. A. Gustafson (1987) Multilateral summation theorems for ordinary and basic hypergeometric series in U ( n ) . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.
  • 9: Bibliography T
  • A. Takemura (1984) Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.
  • I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.
  • N. M. Temme (1993) Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.
  • G. P. Tolstov (1962) Fourier Series. Prentice-Hall Inc., Englewood Cliffs, N.J..
  • J. F. Traub (1964) Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J..
  • 10: Bibliography C
  • B. C. Carlson (1961b) Some series and bounds for incomplete elliptic integrals. J. Math. and Phys. 40, pp. 125–134.
  • B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.
  • H. S. Carslaw (1930) Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • C. W. Clenshaw (1955) A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9 (51), pp. 118–120.