Stirling numbers (first and second kinds)
(0.004 seconds)
11—18 of 18 matching pages
11: 26.15 Permutations: Matrix Notation
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26.15.13
12: Errata
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Table 26.8.1
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Originally the Stirling number was given incorrectly as 6327. The correct number is 63273.
10 |
Reported 2013-11-25 by Svante Janson.
13: Bibliography M
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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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14: Bibliography B
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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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15: 5.11 Asymptotic Expansions
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►For the Bernoulli numbers
, see §24.2(i).
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►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)). … ►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at () and 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
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On the numerical evaluation of the ordinary Bessel function of the second kind.
J. Computational Phys. 21 (3), pp. 343–350.
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Asymptotic estimates of Stirling numbers.
Stud. Appl. Math. 89 (3), pp. 233–243.
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Eigenfunction Expansions Associated with Second-Order Differential Equations.
Clarendon Press, Oxford.
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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations.
Clarendon Press, Oxford.
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Eigenfunction expansions associated with second-order differential equations. Part I.
Second edition, Clarendon Press, Oxford.
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17: Bibliography S
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On the calculation of complex zeros of the modified Bessel function of the second kind.
Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.
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The expansion of Lamé functions into series of associated Legendre functions of the second kind.
Proc. Cambridge Philos. Soc. 62, pp. 441–452.
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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.
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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.
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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
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Algorithm 236: Bessel functions of the first kind.
Comm. ACM 7 (8), pp. 479–480.
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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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Number-Divisor Tables.
British Association Mathematical Tables, Vol. VIII, Cambridge University Press, Cambridge, England.
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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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