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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 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.
  • 7: 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.
  • 8: Errata
  • Changes


    • In Paragraph Inversion Formula in §35.2, the wording was changed to make the integration variable more apparent.

    • In many cases, the links from mathematical symbols to their definitions were corrected or improved. These links were also enhanced with ‘tooltip’ feedback, where supported by the user’s browser.

  • Other Changes


    • In Equations (15.6.1)–(15.6.9), the Olver hypergeometric function F ( a , b ; c ; z ) , previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint | ph ( 1 - z ) | < π has been added. In (15.6.6), the constraint | ph ( - z ) | < π has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when | ph ( 1 - z ) | < π , except (15.6.6) which holds for | ph ( - z ) | < π .”, has been removed.

    • In Subsection 25.2(ii) Other Infinite Series, it is now mentioned that (25.2.5), defines the Stieltjes constants γ n . Consequently, γ n in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

    • In (25.11.36) we have emphasized the link with the Dirichlet L -function, and used the fact that χ ( k ) = 0 . A sentence just below (25.11.36) was added indicating that one should make a comparison with (25.15.1) and (25.15.3).

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

  • Equation (33.11.1)


    33.11.1
    H ± ( η , ρ ) e ± i θ ( η , ρ ) k = 0 ( a ) k ( b ) k k ! ( ± 2 i ρ ) k

    Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an relation.

    Reported by Gergő Nemes on 2019-01-29

  • 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.

  • Subsections 1.15(vi) and 1.15(vii)


    The formulas in these subsections are valid only for x 0 . No conditions on x were given originally.

    Reported 2010-10-18 by Andreas Kurt Richter.