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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: Bibliography O
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Studies on the Painlevé equations. II. Fifth Painlevé equation
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Japan. J. Math. (N.S.) 13 (1), pp. 47–76.
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Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities.
SIAM J. Math. Anal. 8 (4), pp. 673–700.
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Formulae and Tables, The Modified Quotients of Cylinder Functions.
Technical report
Technical Report UDC 517.564.3:518.25, Vol. 4, Report of the Institute of Industrial Science, University of Tokyo, Institute of Industrial Science, Chiba City, Japan.
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6: 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
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Toda equation and its solutions in special functions.
J. Phys. Soc. Japan 65 (6), pp. 1589–1597.
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The Development of Prime Number Theory: From Euclid to Hardy and Littlewood.
Springer-Verlag, Berlin.
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An Introduction to the Theory of Numbers.
5th edition, John Wiley & Sons Inc., New York.
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8: Bibliography Y
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Computation of Kummer functions for large argument by using the -method.
Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).
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9: 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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