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1: Bibliography J
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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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Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent.
Phys. D 1 (1), pp. 80–158.
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Uniform asymptotic expansions for Meixner polynomials.
Constr. Approx. 14 (1), pp. 113–150.
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Note sur la série
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Bull. Soc. Math. France 17, pp. 142–152 (French).
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Memoire sur l’itération des fonctions rationnelles.
J. Math. Pures Appl. 8 (1), pp. 47–245 (French).
2: 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). …3: Bibliography D
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Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2.
American Mathematical Society, Providence, RI.
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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Sums of products of Bernoulli numbers.
J. Number Theory 60 (1), pp. 23–41.
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The Bose-Einstein integrals
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Appl. Sci. Res. B. 6, pp. 240–244.
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The Fermi-Dirac integrals
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Appl. Sci. Res. B. 6, pp. 225–239.
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4: 26.8 Set Partitions: Stirling Numbers
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denotes the Stirling number of the first kind: times the number of permutations of with exactly cycles.
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denotes the Stirling number of the second kind: the number of partitions of into exactly nonempty subsets.
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►where is the Pochhammer symbol: .
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►For ,
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►For asymptotic approximations for and that apply uniformly for as see Temme (1993) and Temme (2015, Chapter 34).
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5: 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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Stirling numbers of the first kind. | |
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6: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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is the number of ways of placing distinct objects into labeled boxes so that there are objects in the th box.
It is also the number of -dimensional lattice paths from to .
For , the multinomial coefficient is defined to be .
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is the number of permutations of with cycles of length 1, cycles of length 2, , and cycles of length :
… is the number of set partitions of with subsets of size 1, subsets of size 2, , and subsets of size :
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7: 24.20 Tables
§24.20 Tables
►Abramowitz and Stegun (1964, Chapter 23) includes exact values of , , ; , , , , 20D; , , 18D. ►Wagstaff (1978) gives complete prime factorizations of and for and , respectively. … ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).8: 22.7 Landen Transformations
9: 26.13 Permutations: Cycle Notation
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denotes the set of permutations of .
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►is in cycle notation.
…In consequence, (26.13.2) can also be written as .
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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.
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