About the Project

Stirling%20numbers

AdvancedHelp

(0.002 seconds)

7 matching pages

1: Bibliography C
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • 2: 26.13 Permutations: Cycle Notation
    The number of elements of 𝔖 n with cycle type ( a 1 , a 2 , , a n ) is given by (26.4.7). The Stirling cycle numbers of the first kind, denoted by [ n k ] , count the number of permutations of { 1 , 2 , , n } with exactly k cycles. They are related to Stirling numbers of the first kind by …See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … The derangement number, d ( n ) , is the number of elements of 𝔖 n with no fixed points: …
    3: 5.11 Asymptotic Expansions
    For the Bernoulli numbers B 2 k , see §24.2(i). … Wrench (1968) gives exact values of g k up to g 20 . …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)). …
    4: 26.14 Permutations: Order Notation
    As an example, 35247816 is an element of 𝔖 8 . The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … The Eulerian number, denoted n k , is the number of permutations in 𝔖 n with exactly k descents. …
    §26.14(iii) Identities
    In this subsection S ( n , k ) is again the Stirling number of the second kind (§26.8), and B m is the m th Bernoulli number24.2(i)). …
    5: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • K. Girstmair (1990b) Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.
  • 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.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • 6: Bibliography M
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • D. S. Moak (1984) The q -analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.
  • C. Mortici (2011a) A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.
  • 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.
  • 7: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • 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.