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21: 27.2 Functions
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►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing .
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►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
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►The
numbers
are relatively prime to and distinct (mod ).
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§27.2(ii) Tables
…22: Bibliography G
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A theorem on the numerators of the Bernoulli numbers.
Amer. Math. Monthly 97 (2), pp. 136–138.
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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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Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point.
Translated from the Russian by J. Shorr-Kon, Office of Technical Services, U. S. Department of Commerce, Washington, D.C..
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Stirling number representation problems.
Proc. Amer. Math. Soc. 11 (3), pp. 447–451.
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23: 24.10 Arithmetic Properties
§24.10 Arithmetic Properties
… ►The denominator of is the product of all these primes . … ►§24.10(ii) Kummer Congruences
… ►§24.10(iii) Voronoi’s Congruence
… ►§24.10(iv) Factors
…24: 26.8 Set Partitions: Stirling Numbers
§26.8 Set Partitions: Stirling Numbers
►§26.8(i) Definitions
► denotes the Stirling number of the first kind: times the number of permutations of with exactly cycles. … ► denotes the Stirling number of the second kind: the number of partitions of into exactly nonempty subsets. … ►§26.8(vi) Relations to Bernoulli Numbers
…25: 24.14 Sums
§24.14 Sums
►§24.14(i) Quadratic Recurrence Relations
… ►§24.14(ii) Higher-Order Recurrence Relations
►In the following two identities, valid for , the sums are taken over all nonnegative integers with . … ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).26: 26.21 Tables
§26.21 Tables
►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients for up to 50 and up to 25; extends Table 26.4.1 to ; tabulates Stirling numbers of the first and second kinds, and , for up to 25 and up to ; tabulates partitions and partitions into distinct parts for up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts , partitions into parts , and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to . ►Goldberg et al. (1976) contains tables of binomial coefficients to and Stirling numbers to .27: 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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28: Bibliography F
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Tablicy značeniĭ funkcii ot kompleksnogo argumenta.
Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).
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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I.
Technical report
Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.
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Faster computation of Bernoulli numbers.
J. Algorithms 13 (3), pp. 431–445.
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An Index of Mathematical Tables. Vols. I, II.
2nd edition, Published for Scientific Computing Service Ltd., London, by
Addison-Wesley Publishing Co., Inc., Reading, MA.
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Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond.
In Recent Perspectives in Random Matrix Theory and Number Theory,
London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.
29: 26.1 Special Notation
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►Other notations for , the Stirling numbers of the first kind, include (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), (Jordan (1939), Moser and Wyman (1958a)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)).
►Other notations for , the Stirling numbers of the second kind, include (Fort (1948)), (Jordan (1939)), (Moser and Wyman (1958b)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).
binomial coefficient. | |
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Eulerian number. | |
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Bell number. | |
Catalan number. | |
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30: Bibliography W
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Prime Divisors of the Bernoulli and Euler Numbers.
In Number Theory for the Millennium, III (Urbana, IL, 2000),
pp. 357–374.
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Generating functions of class-numbers.
Compositio Math. 1, pp. 39–68.
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Rapid approximation to the Voigt/Faddeeva function and its derivatives.
J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.
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Some transformations of generalized hypergeometric series.
Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
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