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1: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: 26.8 Set Partitions: Stirling Numbers
§26.8 Set Partitions: Stirling Numbers
… ► denotes the Stirling number of the first kind: times the number of permutations of with exactly cycles. … … ►Let and be the matrices with th elements , and , respectively. … ►For asymptotic approximations for and that apply uniformly for as see Temme (1993) and Temme (2015, Chapter 34). …3: 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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Bell number. | |
Catalan number. | |
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Stirling numbers of the first kind. | |
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4: 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 .5: 24.15 Related Sequences of Numbers
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§24.15(i) Genocchi Numbers
… ►§24.15(ii) Tangent Numbers
… ►§24.15(iii) Stirling Numbers
►The Stirling numbers of the first kind , and the second kind , are as defined in §26.8(i). … ►§24.15(iv) Fibonacci and Lucas Numbers
…6: 26.13 Permutations: Cycle Notation
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►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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►A permutation is even or odd according to the parity of the number of transpositions.
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7: 24.16 Generalizations
§24.16 Generalizations
… ►Polynomials and Numbers of Integer Order
… ►Bernoulli Numbers of the Second Kind
… ►Degenerate Bernoulli Numbers
… ►Here again denotes the Stirling number of the first kind. …8: Bibliography J
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Classical Electrodynamics.
3rd edition, John Wiley & Sons Inc., New York.
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Fonctions de Mathieu et polynômes de Klein-Gordon.
C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).
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Continuous Univariate Distributions.
2nd edition, Vol. I, John Wiley & Sons Inc., New York.
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Acoustic and Electromagnetic Waves.
Oxford Science Publications, The Clarendon Press Oxford University Press, New York.
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Angular Momentum Theory for Diatomic Molecules.
Academic Press, New York.
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9: Bibliography D
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Multiplicative Number Theory.
3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.
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Interpolation and Approximation.
Dover Publications Inc., New York.
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Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2.
American Mathematical Society, Providence, RI.
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History of the Theory of Numbers (3 volumes).
Carnegie Institution of Washington, Washington, D.C..
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The Taylor Series.
Oxford University Press, Oxford.
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10: Errata
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►This release increments the minor version number and contains considerable additions of new material and clarifications.
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►This release increments the minor version number and contains considerable additions of new material and clarifications.
…These enable insertions of new numbered objects between existing ones without affecting their permanent identifiers and URLs.
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Chapter 25 Zeta and Related Functions
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Table 26.8.1
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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.
Originally the Stirling number was given incorrectly as 6327. The correct number is 63273.
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Reported 2013-11-25 by Svante Janson.