# Gaussian polynomials

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## 1—10 of 25 matching pages

##### 1: 26.21 Tables
It also contains a table of Gaussian polynomials up to $\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}$. …
##### 2: 26.9 Integer Partitions: Restricted Number and Part Size
26.9.4 $\genfrac{[}{]}{0.0pt}{}{m}{n}_{q}=\prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}},$ $n\geq 0$,
is the Gaussian polynomial (or $q$-binomial coefficient); see also §§17.2(i)17.2(ii). …
26.9.5 $\sum_{n=0}^{\infty}p_{k}\left(n\right)q^{n}=\prod_{j=1}^{k}\frac{1}{1-q^{j}}=1% +\sum_{m=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{k+m-1}{m}_{q}q^{m},$
26.9.6 $\sum_{n=0}^{\infty}p_{k}\left(\leq m,n\right)q^{n}=\genfrac{[}{]}{0.0pt}{}{m+k% }{k}_{q}.$
26.9.7 $\sum_{m,n=0}^{\infty}p_{k}\left(\leq m,n\right)x^{k}q^{n}=1+\sum_{k=1}^{\infty% }\genfrac{[}{]}{0.0pt}{}{m+k}{k}_{q}x^{k}=\prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}.$
##### 3: 26.16 Multiset Permutations
The $q$-multinomial coefficient is defined in terms of Gaussian polynomials26.9(ii)) by
26.16.1 $\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q% }=\prod_{k=1}^{n-1}\genfrac{[}{]}{0.0pt}{}{a_{k}+a_{k+1}+\cdots+a_{n}}{a_{k}}_% {q},$
##### 4: 17.2 Calculus
17.2.27 $\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\frac{\left(q;q\right)_{n}}{\left(q;q\right)% _{m}\left(q;q\right)_{n-m}}\\ =\frac{\left(q^{-n};q\right)_{m}(-1)^{m}q^{nm-\genfrac{(}{)}{0.0pt}{}{m}{2}}}{% \left(q;q\right)_{m}},$
17.2.28 $\lim_{q\to 1}\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\genfrac{(}{)}{0.0pt}{}{n}{m}=% \frac{n!}{m!(n-m)!},$
17.2.29 $\genfrac{[}{]}{0.0pt}{}{m+n}{m}_{q}=\frac{\left(q^{n+1};q\right)_{m}}{\left(q;% q\right)_{m}},$
17.2.30 $\genfrac{[}{]}{0.0pt}{}{-n}{m}_{q}=\genfrac{[}{]}{0.0pt}{}{m+n-1}{m}_{q}(-1)^{% m}q^{-mn-\genfrac{(}{)}{0.0pt}{}{m}{2}},$
17.2.31 $\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\genfrac{[}{]}{0.0pt}{}{n-1}{m-1}_{q}+q^{m}% \genfrac{[}{]}{0.0pt}{}{n-1}{m}_{q},$
##### 5: 17.3 $q$-Elementary and $q$-Special Functions
17.3.8 $A_{m,s}\left(q\right)=q^{\genfrac{(}{)}{0.0pt}{}{s-m}{2}+\genfrac{(}{)}{0.0pt}% {}{s}{2}}\sum_{j=0}^{s}(-1)^{j}q^{\genfrac{(}{)}{0.0pt}{}{j}{2}}\genfrac{[}{]}% {0.0pt}{}{m+1}{j}_{q}\frac{(1-q^{s-j})^{m}}{(1-q)^{m}}.$
##### 6: 26.1 Special Notation
 $\genfrac{(}{)}{0.0pt}{}{m}{n}$ binomial coefficient. … Gaussian polynomial. …
##### 7: 26.10 Integer Partitions: Other Restrictions
26.10.3 $(1-x)\sum_{m,n=0}^{\infty}p_{m}\left(\leq k,\mathcal{D},n\right)x^{m}q^{n}=% \sum_{m=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{m}_{q}q^{m(m+1)/2}x^{m}=\prod_{j=1}^{% k}(1+x\,q^{j}),$ $|x|<1$,
##### 8: 18.27 $q$-Hahn Class
18.27.4 $\sum_{y=0}^{N}Q_{n}(q^{-y})Q_{m}(q^{-y})\genfrac{[}{]}{0.0pt}{}{N}{y}_{q}\frac% {\left(\alpha q;q\right)_{y}\left(\beta q;q\right)_{N-y}}{\left(\alpha q\right% )^{y}}=h_{n}\delta_{n,m},$ $n,m=0,1,\ldots,N$,
##### 9: 35.1 Special Notation
Related notations for the Bessel functions are $\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)$ (Terras (1988, pp. 49–64)), and $\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)$ (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).