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Stirling formula

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1: 5.11 Asymptotic Expansions
For explicit formulas for g k in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of g k as k see Boyd (1994) and Nemes (2015a).
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
  • D. S. Moak (1984) The q -analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.
  • 3: Bibliography S
  • R. Spira (1971) Calculation of the gamma function by Stirling’s formula. Math. Comp. 25 (114), pp. 317–322.
  • 4: 8.11 Asymptotic Approximations and Expansions
    This reference also contains explicit formulas for b k ( λ ) in terms of Stirling numbers and for the case λ > 1 an asymptotic expansion for b k ( λ ) as k . … This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. …
    5: 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.
  • 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.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • 6: Bibliography J
  • E. Jahnke and F. Emde (1945) Tables of Functions with Formulae and Curves. 4th edition, Dover Publications, New York.
  • 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.
  • X.-S. Jin and R. Wong (1999) Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.
  • F. Johansson (2012) Efficient implementation of the Hardy-Ramanujan-Rademacher formula. LMS J. Comput. Math. 15, pp. 341–359.
  • 7: 4.13 Lambert W -Function
    4.13.5_2 1 1 + W 0 ( z ) = n = 0 n n n ! z n , | z | < e 1 .
    See Jeffrey and Murdoch (2017) for an explicit representation for the c n in terms of associated Stirling numbers. … For the definition of Stirling cycle numbers of the first kind [ n k ] see (26.13.3). …
    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 ! ,
    8: 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.
  • W. E. Bleick and P. C. C. Wang (1974) Asymptotics of Stirling numbers of the second kind. Proc. Amer. Math. Soc. 42 (2), pp. 575–580.
  • 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.
  • 9: Bibliography C
  • R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.
  • L. Carlitz (1961a) A recurrence formula for ζ ( 2 n ) . Proc. Amer. Math. Soc. 12 (6), pp. 991–992.
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • J. Choi and A. K. Rathie (2013) An extension of a Kummer’s quadratic transformation formula with an application. Proc. Jangjeon Math. Soc. 16 (2), pp. 229–235.
  • R. Cools (2003) An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.
  • 10: Errata
  • Subsection 17.9(iii)

    The title of the paragraph which was previously “Gasper’s q -Analog of Clausen’s Formula” has been changed to “Gasper’s q -Analog of Clausen’s Formula (16.12.2)”.

  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • Usability

    Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the [Uncaptioned image] icons to the right of each formula, and provide better search capabilities.

  • Table 26.8.1

    Originally the Stirling number s ( 10 , 6 ) was given incorrectly as 6327. The correct number is 63273.

    n k
    0 1 2 3 4 5 6 7 8 9 10
    10 0 3 62880 10 26576 11 72700 7 23680 2 69325 63273 9450 870 45 1

    Reported 2013-11-25 by Svante Janson.

  • Subsection 14.18(iii)

    This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.