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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 M
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On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade.
Funkcial. Ekvac. 46 (1), pp. 121–171.
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The Calculus of Finite Differences.
Macmillan and Co. Ltd., London.
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Methods of Theoretical Physics.
Vol. 1, McGraw-Hill Book Co., New York.
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Methods of Theoretical Physics.
Vol. 2, McGraw-Hill Book Co., New York.
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Further improvements of some double inequalities for bounding the gamma function.
Math. Comput. Modelling 57 (5-6), pp. 1360–1363.
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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
…7: 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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8: 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. …9: Mourad E. H. Ismail
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►His well-known book Classical and Quantum Orthogonal Polynomials in One Variable was published by Cambridge University Press in 2005 and reprinted with corrections in paperback in Ismail (2009).
►Books for which he has served as editor or coeditor include, Mathematical Analysis, Wavelets, and Signal Processing (with M.
… Suslov), Kluwer Academic Publishers, 2001; Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (with F.
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►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive
Approximation; Journal of Approximation Theory; and Integral Transforms
and Special Functions.
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10: Antony Ross Barnett
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►Barnett’s research interests included number theory and special functions.
He is coauthor of the book Computing for Scientists (with R.
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