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11: 26.10 Integer Partitions: Other Restrictions

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§26.10(ii) Generating Functions

… ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►

26.10.10 $$p(𝒟k,n)=\sum p_m\left(n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\right),$$

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Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(iii), §26.10 and Ch.26

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§26.10(vi) Bessel-Function Expansion

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26.10.17 $$p(𝒟,n)=\pi \sum _{k=1}^{\mathrm{\infty }}\frac{A_{2k-1}(n)}{(2k-1)\sqrt{24n+1}}{I}_1\left(\frac{\pi }{2k-1}\sqrt{\frac{24n+1}{72}}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $k$: nonnegative integer, $n$: nonnegative integer and $A_k(n)$: coefficient

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(vi), §26.10 and Ch.26

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12: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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13: 27.13 Functions

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§27.13(i) Introduction

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14: Bibliography R

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On the partition function p(n). Proc. London Math. Soc. (2) 43 (4), pp. 241–254.

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External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(iii).

Encodings:

BibTeX

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15: Bibliography O

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Distribution of the partition function modulo $m$ . Ann. of Math. (2) 151 (1), pp. 293–307.

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External Links:

ISSN 0003-486X, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§27.14(v).

Encodings:

BibTeX

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16: 26.12 Plane Partitions

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§26.12(ii) Generating Functions

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26.12.26 $$\mathrm{pp}\left(n\right)\sim \frac{{\left(\zeta \left(3\right)\right)}^{7/36}}{2^{11/36}{(3\pi )}^{1/2}{n}^{25/36}}\exp \left(3{\left(\zeta \left(3\right)\right)}^{1/3}{\left(\frac{1}{2}n\right)}^{2/3}+{\zeta }^{\prime }\left(-1\right)\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\sim$: asymptotic equality, $\exp z$: exponential function, $\mathrm{pp}\left(n\right)$: 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, $\mathrm{pp}\left(n\right)\sim {\left(\frac{\zeta \left(3\right)}{2^{11}{n}^{25}}\right)}^{1/36}\exp \left(3{\left(\frac{\zeta \left(3\right)n^2}{4}\right)}^{1/3}+{\zeta }^{\prime }\left(-1\right)\right)$, 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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17: Bibliography L

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A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

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External Links:

Document, MathReview (R. D. James), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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18: Bibliography C

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Computing the integer partition function. Math. Comp. 76 (259), pp. 1619–1638.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (Donald L. Vestal, Jr.), ZentralBlatt

Referenced by:

§27.20.

Encodings:

BibTeX

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19: Bibliography

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On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.

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External Links:

Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§17.6(ii).

Encodings:

BibTeX

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20: 3.7 Ordinary Differential Equations

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3.7.11 $$𝐰={[w(z_0),w^{\prime }(z_0),w(z_1),w^{\prime }(z_1),\mathrm{\dots },w(z_P),w^{\prime }(z_P)]}^{\mathrm{T}},$$

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Symbols:

$w(z)$: function and $P$: partitioning point

Permalink:

http://dlmf.nist.gov/3.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(iii), §3.7 and Ch.3

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