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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: 34.6 Definition: Symbol
§34.6 Definition: Symbol
►The symbol may be defined either in terms of symbols or equivalently in terms of symbols: ►
34.6.1
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34.6.2
►The symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments.
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3: 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). …4: 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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5: 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 .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
…7: 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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On an asymptotic expansion of a ratio of gamma functions.
Proc. Roy. Irish Acad. Sect. A 95 (1), pp. 5–9.
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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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Solution of Equations in Euclidean and Banach Spaces.
Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London.
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8: Bibliography N
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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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9: 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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