Stirling%20numbers
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1: Bibliography C
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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2: 26.13 Permutations: Cycle Notation
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►The number of elements of with cycle type is given by (26.4.7).
►The Stirling cycle numbers of the first kind, denoted by , count the number of permutations of with exactly cycles.
They are related to Stirling numbers of the first kind by
…See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►The derangement number, , is the number of elements of with no fixed points:
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3: 5.11 Asymptotic Expansions
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►For the Bernoulli numbers
, see §24.2(i).
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►Wrench (1968) gives exact values of up to .
…For explicit formulas for in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of as see Boyd (1994) and Nemes (2015a).
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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
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►As an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
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►The Eulerian number, denoted , is the number of permutations in with exactly descents.
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§26.14(iii) Identities
►In this subsection is again the Stirling number of the second kind (§26.8), and is the th Bernoulli number (§24.2(i)). …5: Bibliography G
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Dirichlet convolution of cotangent numbers and relative class number formulas.
Monatsh. Math. 110 (3-4), pp. 231–256.
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Tables of binomial coefficients and Stirling numbers.
J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.
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Stirling number representation problems.
Proc. Amer. Math. Soc. 11 (3), pp. 447–451.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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6: Bibliography M
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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The -analogue of Stirling’s formula.
Rocky Mountain J. Math. 14 (2), pp. 403–413.
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A new Stirling series as continued fraction.
Numer. Algorithms 56 (1), pp. 17–26.
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Asymptotic development of the Stirling numbers of the first kind.
J. London Math. Soc. 33, pp. 133–146.
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Stirling numbers of the second kind.
Duke Math. J. 25 (1), pp. 29–43.
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7: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Asymptotics of Stirling numbers of the second kind.
Proc. Amer. Math. Soc. 42 (2), pp. 575–580.
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