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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: 26.4 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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5: 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).6: 22.7 Landen Transformations
7: 26.3 Lattice Paths: Binomial Coefficients
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is the number of ways of choosing objects from a collection of distinct objects without regard to order.
is the number of lattice paths from to .
…The number of lattice paths from to , , that stay on or above the line is
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26.3.4
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26.3.6
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8: 24.6 Explicit Formulas
§24.6 Explicit Formulas
►The identities in this section hold for . … ►
24.6.1
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24.6.4
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24.6.10
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