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11: 20 Theta Functions

Chapter 20 Theta Functions

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12: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

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§35.4(i) Definitions

►A partition $\kappa =(k_1,\mathrm{\dots },k_m)$ is a vector of nonnegative integers, listed in nonincreasing order. Also, $|\kappa |$ denotes $k_1+\mathrm{\cdots }+k_m$, the weight of $\kappa$; $\mathrm{\ell }(\kappa )$ denotes the number of nonzero $k_j$; $a+\kappa$ denotes the vector $(a+k_1,\mathrm{\dots },a+k_m)$. ►The partitional shifted factorial is given by …

13: 5.20 Physical Applications

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Rutherford Scattering

… ►Then the partition function (with $\beta =1/(kT)$) is given by … ►and the partition function is given by … ►

Elementary Particles

… ►Carlitz (1972) describes the partition function of dense hadronic matter in terms of a gamma function. …

14: 26.20 Physical Applications

… ►The latter reference also describes chemical applications of other combinatorial techniques. ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). …

15: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function $p\left(n\right)$ for . …To compute a particular value $p\left(n\right)$ it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). …

16: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

Encodings:

BibTeX

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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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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Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.

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External Links:

ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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Plane partitions. III. The weak Macdonald conjecture. Invent. Math. 53 (3), pp. 193–225.

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External Links:

ISSN 0020-9910, Document, MathReview (Edward A. Bender), ZentralBlatt

Referenced by:

§26.12(i), §26.12(ii).

Encodings:

BibTeX

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17: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

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26.8.3 $n>k\ge 1$.

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Symbols:

$s(n,k)$: Stirling number of the first kind, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(i), §26.8 and Ch.26

► $S(n,k)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\mathrm{\dots },n\}$ into exactly $k$ nonempty subsets. … ►

26.8.7 $$\sum _{k=0}^{n}s(n,k)x^k={\left(x-n+1\right)}_n,$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $s(n,k)$: Stirling number of the first kind, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.3 (in slightly different form)

Permalink:

http://dlmf.nist.gov/26.8.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §26.8(ii), §26.8 and Ch.26

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18: 8 Incomplete Gamma and Related
Functions

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19: 28 Mathieu Functions and Hill’s Equation

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20: 17.16 Mathematical Applications

… ►Many special cases of $q$-series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9. …