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Stirling numbers (first and second kinds)

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11: 26.15 Permutations: Matrix Notation
12: Errata
  • 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.

  • 13: Bibliography M
  • L. Moser and M. Wyman (1958a) Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.
  • L. Moser and M. Wyman (1958b) Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.
  • 14: Bibliography B
  • 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.
  • 15: 5.11 Asymptotic Expansions
    For the Bernoulli numbers B 2 k , see §24.2(i). … 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)). … If the sums in the expansions (5.11.1) and (5.11.2) are terminated at k = n 1 ( k 0 ) and z is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. …
    16: Bibliography T
  • N. M. Temme (1976) On the numerical evaluation of the ordinary Bessel function of the second kind. J. Computational Phys. 21 (3), pp. 343–350.
  • N. M. Temme (1993) Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.
  • E. C. Titchmarsh (1946) Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.
  • E. C. Titchmarsh (1958) Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.
  • E. C. Titchmarsh (1962a) Eigenfunction expansions associated with second-order differential equations. Part I. Second edition, Clarendon Press, Oxford.
  • 17: Bibliography S
  • S. L. Skorokhodov (1985) On the calculation of complex zeros of the modified Bessel function of the second kind. Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.
  • B. D. Sleeman (1966b) The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.
  • R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.
  • R. Szmytkowski (2011) On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.
  • R. Szmytkowski (2012) On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.
  • 18: Bibliography G
  • W. Gautschi (1964b) Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.
  • K. Girstmair (1990b) Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.
  • J. W. L. Glaisher (1940) Number-Divisor Tables. British Association Mathematical Tables, Vol. VIII, Cambridge University Press, Cambridge, England.
  • 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.