pentagonal numbers

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1: 27.14 Unrestricted Partitions

… ►Euler’s pentagonal number theorem states that ►

27.14.4 $$f\left(x\right)=1-x-x^2+x^5+x^7-x^{12}-x^{15}+\mathrm{\cdots }=1+\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k\left(x^{\omega (k)}+x^{\omega (-k)}\right),$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $k$: positive integer, $x$: real number and $\omega (k)$: pentagonal numbers

Permalink:

http://dlmf.nist.gov/27.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

►where the exponents $1$, $2$, $5$, $7$, $12$, $15$, $\mathrm{\dots }$ are the pentagonal numbers, defined by ►

27.14.5 $$\omega (\pm k)=(3k^2\mp k)/2,$$ $k=1,2,3,\mathrm{\dots }$.

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Defines:

$\omega (k)$: pentagonal numbers (locally)

Symbols:

$k$: positive integer

Permalink:

http://dlmf.nist.gov/27.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

… ►

27.14.6 $$p\left(n\right)=\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\left(p\left(n-\omega (k)\right)+p\left(n-\omega (-k)\right)\right)=p\left(n-1\right)+p\left(n-2\right)-p\left(n-5\right)-p\left(n-7\right)+\mathrm{\cdots },$$

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Symbols:

$p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer, $n$: positive integer and $\omega (k)$: pentagonal numbers

A&S Ref:

24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

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2: 20.12 Mathematical Applications

… ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s $\tau \left(n\right)$ function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …

3: Bibliography

… ►

Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.

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Notes:

In memory of Gian-Carlo Rota

External Links:

ISSN 0097-3165, Document, MathReview (Alessandro Di Bucchianico), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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