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11: 28.6 Expansions for Small
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►For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).
►The coefficients of the power series of , and also , are the same until the terms in and , respectively.
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►Here for , for , and for and .
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►where is the unique root of the equation in the interval , and .
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►For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).
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12: 25.20 Approximations
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of and , , for (23D).
13: 24.20 Tables
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►Abramowitz and Stegun (1964, Chapter 23) includes exact values of , , ; , , , , 20D; , , 18D.
►Wagstaff (1978) gives complete prime factorizations of and for and , respectively.
In Wagstaff (2002) these results are extended to and , respectively, with further complete and partial factorizations listed up to 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).
14: 27.2 Functions
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►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing .
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►The numbers are relatively prime to and distinct (mod ).
…It is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
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27.2.12
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15: Bibliography
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Asymptotic expansions of spheroidal wave functions.
J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.
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Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable.
2nd edition, McGraw-Hill Book Co., New York.
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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations.
Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
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Normal forms of functions in the neighborhood of degenerate critical points.
Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
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Some basic hypergeometric extensions of integrals of Selberg and Andrews.
SIAM J. Math. Anal. 11 (6), pp. 938–951.
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16: Bibliography E
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The penetration of a potential barrier by electrons.
Phys. Rev. 35 (11), pp. 1303–1309.
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Interlacing properties of the zeros of Bessel functions.
Atti Sem. Mat. Fis. Univ. Modena XLII (2), pp. 525–529.
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Higher Transcendental Functions. Vol. II.
McGraw-Hill Book Company, Inc., New York-Toronto-London.
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Painlevé transcendent describes quantum correlation function of the antiferromagnet away from the free-fermion point.
J. Phys. A 29 (17), pp. 5619–5626.
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Institutiones Calculi Integralis.
Opera Omnia (1), Vol. 11, pp. 110–113.
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17: Bibliography L
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Eine Verallgemeinerung der Sphäroidfunktionen.
Arch. Math. 11, pp. 29–39.
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Note sur la fonction
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Acta Math. 11 (1-4), pp. 19–24 (French).
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Algorithm 244: Fresnel integrals.
Comm. ACM 7 (11), pp. 660–661.
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On the theory of Painlevé’s third equation.
Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).
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Algorithms for rational approximations for a confluent hypergeometric function.
Utilitas Math. 11, pp. 123–151.
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18: 26.12 Plane Partitions
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►The number of self-complementary plane partitions in is
…in it is
…in it is
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►The number of symmetric self-complementary plane partitions in is
…in it is
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19: 18.8 Differential Equations
20: Bibliography K
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Bernstein, Pick, Poisson and related integral expressions for Lambert
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Integral Transforms Spec. Funct. 23 (11), pp. 817–829.
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Auxiliary table for the incomplete elliptic integrals.
J. Math. Physics 27, pp. 11–36.
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An extension of Saalschütz’s summation theorem for the series
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Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
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Algorithm 327: Dilogarithm [S22].
Comm. ACM 11 (4), pp. 270–271.
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Clebsch-Gordan coefficients for and Hahn polynomials.
Nieuw Arch. Wisk. (3) 29 (2), pp. 140–155.
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