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21: 26.12 Plane Partitions

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Table 26.12.1: Plane partitions.

$n$	$\operatorname{pp}\left(n\right)$	$n$	$\operatorname{pp}\left(n\right)$	$n$	$\operatorname{pp}\left(n\right)$
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10	500	27	16 32658	44	12343 63833
11	859	28	24 83234	45	17740 79109
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13	2485	30	56 68963	47	36379 93036
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16	11297	33	189 74973	50	1 04996 40707

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26.12.26

\operatorname{pp}\left(n\right)\sim\frac{\left(\zeta\left(3\right)\right)^{7/3% 6}}{2^{11/36}(3\pi)^{1/2}n^{25/36}}\exp\left(3\left(\zeta\left(3\right)\right)% ^{1/3}\left(\tfrac{1}{2}n\right)^{2/3}+\zeta'\left(-1\right)\right),

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\sim$ : asymptotic equality, $\exp\NVar{z}$ : exponential function, $\operatorname{pp}\left(\NVar{n}\right)$ : number of plane partitions of $n$ , $n$ : nonnegative integer and $\pi$ : plane partition
Referenced by:: §26.12(iv), Erratum (V1.0.5) for Equation (26.12.26)
Permalink:: http://dlmf.nist.gov/26.12.E26
Encodings:: pMML, png, TeX
Errata (effective with 1.0.5):: The original statement of this equation, $\operatorname{pp}\left(n\right)\sim\left(\frac{\zeta\left(3\right)}{2^{11}n^{2% 5}}\right)^{1/36}\*\exp\left(3\left(\frac{\zeta\left(3\right)n^{2}}{4}\right)^% {1/3}+\zeta'\left(-1\right)\right)$ , has been corrected.
Reported 2011-09-05 by Suresh Govindarajan
See also:: Annotations for §26.12(iv), §26.12 and Ch.26

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22: Bibliography S

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R. Shail (1980) On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.

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External Links:: ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt
Referenced by:: §29.17(i), §29.8.
Encodings:: BibTeX

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N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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External Links:: ISSN 0253-424X, MathReview
Referenced by:: §9.10(v).
Encodings:: BibTeX

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A. Sidi (2010) A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

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External Links:: ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §2.3(iv).
Encodings:: BibTeX

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R. Sips (1965) Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill. Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.

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External Links:: ZentralBlatt
Referenced by:: §28.8(ii).
Encodings:: BibTeX

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K. Soni (1980) Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.

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External Links:: ISSN 0036-1410, Document, MathReview (John S. Lew), ZentralBlatt
Referenced by:: §2.5(ii).
Encodings:: BibTeX

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23: 10.75 Tables

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Wills et al. (1982) tabulates $j_{0,m}$ , $j_{1,m}$ , $y_{0,m}$ , $y_{1,m}$ for $m=1(1)30$ , 35D.

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MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function $J_{0}\left(z\right)-iJ_{1}\left(z\right)$ , 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).

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Abramowitz and Stegun (1964, Chapter 11) tabulates $\int_{0}^{x}J_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 10D; $\int_{0}^{x}t^{-1}(1-J_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 8D.

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Abramowitz and Stegun (1964, Chapter 11) tabulates $e^{-x}\int_{0}^{x}I_{0}\left(t\right)\,\mathrm{d}t$ , $e^{x}\int_{x}^{\infty}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 7D; $e^{-x}\int_{0}^{x}t^{-1}(I_{0}\left(t\right)-1)\,\mathrm{d}t$ , $xe^{x}\int_{x}^{\infty}t^{-1}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 6D.

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Zhang and Jin (1996, pp. 296–305) tabulates $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{j}_{n}'\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ , $\mathsf{y}_{n}'\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $\mathsf{k}_{n}'\left(x\right)$ , $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; $x\mathsf{j}_{n}\left(x\right)$ , $(x\mathsf{j}_{n}\left(x\right))^{\prime}$ , $x\mathsf{y}_{n}\left(x\right)$ , $(x\mathsf{y}_{n}\left(x\right))^{\prime}$ (Riccati–Bessel functions and their derivatives), $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; real and imaginary parts of $\mathsf{j}_{n}\left(z\right)$ , $\mathsf{j}_{n}'\left(z\right)$ , $\mathsf{y}_{n}\left(z\right)$ , $\mathsf{y}_{n}'\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(z\right)$ , $\mathsf{k}_{n}\left(z\right)$ , $\mathsf{k}_{n}'\left(z\right)$ , $n=0(1)15$ , 20(10)50, 100, $z=4+2i$ , $20+10i$ , 8S. (For the notation replace $j, y, i, k$ by $\mathsf{j}$ , $\mathsf{y}$ , ${\mathsf{i}^{(1)}}$ , $\mathsf{k}$ , respectively.)

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24: Bibliography N

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National Physical Laboratory (1961) Modern Computing Methods. 2nd edition, Notes on Applied Science, No. 16, Her Majesty’s Stationery Office, London.

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External Links:: MathReview (A. S. Householder), ZentralBlatt
Referenced by:: §3.8(iv).
Encodings:: BibTeX

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G. Nemes (2017a) Error bounds for the asymptotic expansion of the Hurwitz zeta function. Proc. A. 473 (2203), pp. 20170363, 16.

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External Links:: ISSN 1364-5021, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.15.
Encodings:: BibTeX

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E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:: Includes Algol program.
External Links:: MathReview
Referenced by:: §19.39(iii).
Encodings:: BibTeX

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E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

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V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:: In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.
External Links:: ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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25: Bibliography L

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D. Le (1985) An efficient derivative-free method for solving nonlinear equations. ACM Trans. Math. Software 11 (3), pp. 250–262.

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Notes:: Corrigendum, ibid. 15(3) (1989), 287
External Links:: ISSN 0098-3500, Document, MathReview (T. Feagin), ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

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A. Leitner and J. Meixner (1960) Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

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External Links:: ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §30.12.
Encodings:: BibTeX

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H. Lotsch and M. Gray (1964) Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

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Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iv).
Encodings:: BibTeX

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N. A. Lukaševič (1967b) On the theory of Painlevé’s third equation. Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).

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Notes:: In Russian. English translation: Differential Equations 3(11), pp. 994–999
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.10(iii), §32.8(iii), §32.9(i), §32.9(ii).
Encodings:: BibTeX

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Y. L. Luke (1977a) Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.

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External Links:: MathReview (F. Gotze), ZentralBlatt
Referenced by:: §13.31(iii).
Encodings:: BibTeX

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26: Bibliography C

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L. Carlitz (1960) Note on Nörlund’s polynomial

B_{n}^{(z)}

. Proc. Amer. Math. Soc. 11 (3), pp. 452–455.

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External Links:: FullText, MathReview (M. P. Drazin), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

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P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

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J. A. Cochran (1963) Further formulas for calculating approximate values of the zeros of certain combinations of Bessel functions. IEEE Trans. Microwave Theory Tech. 11 (6), pp. 546–547.

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External Links:: ISSN 0018-9480, FullText
Referenced by:: §10.21(x).
Encodings:: BibTeX

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M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

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F. Cooper, A. Khare, and A. Saxena (2006) Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.

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External Links:: ISSN 1076-2787, Document, MathReview
Referenced by:: §19.6(i).
Encodings:: BibTeX

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27: Bibliography D

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P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou (1999b) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.

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External Links:: ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

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S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

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E. Dorrer (1968) Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.

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Notes:: Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39
External Links:: ISSN 0001-0782, Document
Referenced by:: §8.28(iv).
Encodings:: BibTeX

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B. A. Dubrovin (1981) Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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Notes:: With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)
External Links:: ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.
Encodings:: BibTeX

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J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

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External Links:: ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt
Referenced by:: §8.17(i).
Encodings:: BibTeX

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28: Bibliography K

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G. A. Kalugin, D. J. Jeffrey, and R. M. Corless (2012) Bernstein, Pick, Poisson and related integral expressions for Lambert

W

. Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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External Links:: ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

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E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §19.12.
Encodings:: BibTeX

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R. P. Kerr (1963) Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.

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External Links:: Document, MathReview (C. Gilbert), ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

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K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

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Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §25.21(v).
Encodings:: BibTeX

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T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §15.14, §15.14.
Encodings:: BibTeX

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29: Software Index

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	Open Source	With Book	Commercial
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11 Struve and Related Functions
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16 Generalized Hypergeometric Functions & Meijer G-Function
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30 Spheroidal Wave Functions
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