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11: 26.2 Basic Definitions
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►Thus is the permutation , , .
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►Here , and .
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►A lattice path is a directed path on the plane integer lattice .
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►As an example, , , is a partition of .
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►As an example, is a partition of 13.
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12: 26.21 Tables
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►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 .
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13: 26.10 Integer Partitions: Other Restrictions
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►The set is denoted by .
If more than one restriction applies, then the restrictions are separated by commas, for example, .
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►where the sum is over nonnegative integer values of for which .
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►where the sum is over nonnegative integer values of for which .
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►where the sum is over nonnegative integer values of for which .
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14: 28.16 Asymptotic Expansions for Large
15: Publications
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D. W. Lozier, B. R. Miller and B. V. Saunders (1999)
Design of a Digital Mathematical Library for Science, Technology and Education,
Proceedings of the
IEEE Forum on Research and Technology Advances in Digital Libraries (IEEE ADL ’99,
Baltimore, Maryland, May 19, 1999).
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D. W. Lozier (2003)
The NIST Digital Library of Mathematical Functions Project,
Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management,
Vol. 38, Nos. 1–3, pp. 105–119.
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B. R. Miller and A. Youssef (2003)
Technical Aspects of the Digital Library of Mathematical Functions,
Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management,
Vol. 38, Nos. 1–3, pp. 121–136.
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B. V. Saunders and Q. Wang (2006)
From B-Spline Mesh Generation to Effective Visualizations for the
NIST Digital Library of Mathematical Functions,
in Curve and Surface Design, Proceedings of the Sixth International
Conference on Curves and Surfaces,
Avignon, France June 29–July 5, 2006,
pp. 235–243.
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B. I. Schneider, B. R. Miller and B. V. Saunders (2018)
NIST’s Digital Library of Mathematial Functions,
Physics Today
71, 2, 48 (2018), pp. 48–53.
16: 5.17 Barnes’ -Function (Double Gamma Function)
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5.17.2
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5.17.3
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5.17.5
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►Here is the Bernoulli number (§24.2(i)), and is Glaisher’s constant, given by
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5.17.7
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17: Bibliography Z
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Algorithm 916: computing the Faddeyeva and Voigt functions.
ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.
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The Dilogarithm Function in Geometry and Number Theory.
In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.),
Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.
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“Hidden symmetry” of Askey-Wilson polynomials.
Theoret. and Math. Phys. 89 (2), pp. 1146–1157.
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The computation of the smaller coefficients of
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Bull. Amer. Math. Soc. 45 (12), pp. 917–919.
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Tables of the Legendre functions . Part II.
Translated by Prasenjit Basu. Mathematical Tables Series, Vol.
38. A Pergamon Press Book, The Macmillan Co., New York.
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18: Bibliography G
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Algorithm 471: Exponential integrals.
Comm. ACM 16 (12), pp. 761–763.
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Mémoire sur les fonctions hypergéométriques d’ordre supérieur.
Ann. Sci. École Norm. Sup. (2) 12, pp. 261–286, 395–430 (French).
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The solutions of Painlevé’s fifth equation.
Differ. Uravn. 12 (4), pp. 740–742 (Russian).
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One-parameter systems of solutions of Painlevé equations.
Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).
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Algorithm 300: Coulomb wave functions.
Comm. ACM 10 (4), pp. 244–245.
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19: 3.4 Differentiation
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►If is continuous on the interval defined in §3.3(i), then the remainder in (3.4.1) is given by
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►With the choice (which is crucial when is large because of numerical cancellation) the integrand equals at the dominant points , and in combination with the factor in front of the integral sign this gives a rough approximation to .
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►For additional formulas involving values of and on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546).
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