q-binomial%20coefficient

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1: 26.10 Integer Partitions: Other Restrictions

… ►Throughout this subsection it is assumed that . ►

26.10.2 $$\sum _{n=0}^{\mathrm{\infty }}p(𝒟,n)q^n=\prod _{j=1}^{\mathrm{\infty }}(1+q^j)=\prod _{j=1}^{\mathrm{\infty }}\frac{1}{1-q^{2j-1}}=1+\sum _{m=1}^{\mathrm{\infty }}\frac{q^{m(m+1)/2}}{(1-q)(1-q^2)\mathrm{\cdots }(1-q^m)}=1+\sum _{m=1}^{\mathrm{\infty }}{q}^m(1+q)(1+q^2)\mathrm{\cdots }(1+q^{m-1}),$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E2

Encodings:

pMML, png, TeX

See also:

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

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26.10.3 $$(1-x)\sum _{m,n=0}^{\mathrm{\infty }}{p}_m(\le k,𝒟,n)x^m{q}^n=\sum _{m=0}^k{\left[\genfrac{}{}{0.0pt}{}{k}{m}\right]}_q{q}^{m(m+1)/2}{x}^m=\prod _{j=1}^k(1+x{q}^j),$$ ,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Referenced by:

§26.10(ii)

Permalink:

http://dlmf.nist.gov/26.10.E3

Encodings:

pMML, png, TeX

See also:

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

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26.10.4 $$\sum _{n=0}^{\mathrm{\infty }}p(𝒟k,n)q^n=1+\sum _{m=1}^{\mathrm{\infty }}\frac{q^{(k{m}^2+(2-k)m)/2}}{(1-q)(1-q^2)\mathrm{\cdots }(1-q^m)},$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.10.E4

Encodings:

pMML, png, TeX

See also:

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

… ►where $I_1\left(x\right)$ is the modified Bessel function (§10.25(ii)), and …

2: 26.9 Integer Partitions: Restricted Number and Part Size

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26.9.4 $${\left[\genfrac{}{}{0.0pt}{}{m}{n}\right]}_q=\prod _{j=1}^n\frac{1-q^{m-n+j}}{1-q^j},$$ $n\ge 0$,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

►is the Gaussian polynomial (or $q$-binomial coefficient); see also §§17.2(i)–17.2(ii). …It is also assumed everywhere that . … ►

26.9.6 $$\sum _{n=0}^{\mathrm{\infty }}{p}_k(\le m,n)q^n={\left[\genfrac{}{}{0.0pt}{}{m+k}{k}\right]}_q.$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

►Also, when …

3: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

… ►

Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

… ►

The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

BibTeX

… ►

HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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