Gaussian polynomials

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1: 26.21 Tables

… ►It also contains a table of Gaussian polynomials up to ${\left[\genfrac{}{}{0.0pt}{}{12}{6}\right]}_q$. …

2: 26.9 Integer Partitions: Restricted Number and Part Size

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

ⓘ

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). … ►

26.9.5 $$\sum _{n=0}^{\mathrm{\infty }}{p}_k\left(n\right)q^n=\prod _{j=1}^k\frac{1}{1-q^j}=1+\sum _{m=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{k+m-1}{m}\right]}_q{q}^m,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/26.9.E5

Encodings:

pMML, png, TeX

See also:

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

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

ⓘ

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

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26.9.7 $$\sum _{m,n=0}^{\mathrm{\infty }}{p}_k(\le m,n)x^k{q}^n=1+\sum _{k=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{m+k}{k}\right]}_q{x}^k=\prod _{j=0}^m\frac{1}{1-x{q}^j}.$$

ⓘ

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

Permalink:

http://dlmf.nist.gov/26.9.E7

Encodings:

pMML, png, TeX

See also:

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

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3: 26.16 Multiset Permutations

… ►The $q$-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by ►

26.16.1 $${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q=\prod _{k=1}^{n-1}{\left[\genfrac{}{}{0.0pt}{}{a_k+a_{k+1}+\mathrm{\cdots }+a_n}{a_k}\right]}_q,$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), ${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q$: $q$-multinomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer, $a_j$: numbers and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.16 and Ch.26

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4: 17.2 Calculus

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17.2.27 $${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q=\frac{{(q;q)}_n}{{(q;q)}_m{(q;q)}_{n-m}}=\frac{{(q^{-n};q)}_m{(-1)}^m{q}^{nm-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)}}{{(q;q)}_m},$$

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Defines:

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

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

►

17.2.28 $$\underset{q\to 1}{lim}{\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q=\left(\genfrac{}{}{0.0pt}{}{n}{m}\right)=\frac{n!}{m!(n-m)!},$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

►

17.2.29 $${\left[\genfrac{}{}{0.0pt}{}{m+n}{m}\right]}_q=\frac{{(q^{n+1};q)}_m}{{(q;q)}_m},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

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17.2.30 $${\left[\genfrac{}{}{0.0pt}{}{-n}{m}\right]}_q={\left[\genfrac{}{}{0.0pt}{}{m+n-1}{m}\right]}_q{(-1)}^m{q}^{-mn-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)},$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

►

17.2.31 $${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q={\left[\genfrac{}{}{0.0pt}{}{n-1}{m-1}\right]}_q+q^m{\left[\genfrac{}{}{0.0pt}{}{n-1}{m}\right]}_q,$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

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5: 17.3 $q$-Elementary and $q$-Special Functions

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17.3.8 $$A_{m,s}\left(q\right)=q^{\left(\genfrac{}{}{0.0pt}{}{s-m}{2}\right)+\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{m+1}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$

ⓘ

Defines:

$A_{m,s}\left(q\right)$: $q$-Euler number

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

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17.3.9 $$a_{m,s}\left(q\right)=\frac{q^{-\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}{(1-q)}^s}{{(q;q)}_s}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{s}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$

ⓘ

Defines:

$a_{m,s}\left(q\right)$: $q$-Stirling number

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $m$: nonnegative integer and $s$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

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6: 26.1 Special Notation

… ► ►►►

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
…
${\left[\genfrac{}{}{0.0pt}{}{m}{n}\right]}_q$	Gaussian polynomial.
…

…

7: 26.10 Integer Partitions: Other Restrictions

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

ⓘ

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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8: 18.27 $q$-Hahn Class

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18.27.4 $$\sum _{y=0}^N{Q}_n(q^{-y})Q_m(q^{-y}){\left[\genfrac{}{}{0.0pt}{}{N}{y}\right]}_q\frac{{(\alpha q;q)}_y{(\beta q;q)}_{N-y}}{{\left(\alpha q\right)}^y}=h_n{\delta }_{n,m},$$ $n,m=0,1,\mathrm{\dots },N$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $y$: real variable, $N$: positive integer, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Notes:

This is Ismail (2009, (18.5.2)) but the presentation has been simplified.

Referenced by:

§18.27(ii), Erratum (V1.2.0) for Equation (18.27.4)

Permalink:

http://dlmf.nist.gov/18.27.E4

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

Annotations for §18.27(ii), §18.27 and Ch.18

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18.27.4_1 $$h_n=\frac{{\left(\alpha q\right)}^{nN}}{1-\alpha \beta q^{2n+1}}\frac{{(\alpha \beta q^{n+1};q)}_{N+1}{(\beta q;q)}_n}{{\left[\genfrac{}{}{0.0pt}{}{N}{n}\right]}_q{(\alpha q;q)}_n}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $h_n$

Referenced by:

§18.27(ii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(ii), §18.27 and Ch.18

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9: 35.1 Special Notation

… ►Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).

10: 35.9 Applications

§35.9 Applications

… ►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. ►In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).