Gaussian polynomials

For an indeterminate $q$ and integers $n\ge m\ge 0$ we define the following:

(a) ${(m)}_q=q^{m-1}+q^{m-2}+\mathrm{\cdots }+1$ for $m>0$,

(b) ${(m!)}_q={(m)}_q{(m-1)}_q\mathrm{\cdots }{(1)}_q$ for $m>0$, and ${(0!)}_q=1$,

(c) ${\left(\genfrac{}{}{0pt}{}{n}{m}\right)}_q=\frac{{(n!)}_q}{{(m!)}_q{((n-m)!)}_q}$. If $m>n$ then we define ${\left(\genfrac{}{}{0pt}{}{n}{m}\right)}_q=0$.

The expressions ${\left(\genfrac{}{}{0pt}{}{n}{m}\right)}_q$ are called $q$-binomial coefficients or Gaussian polynomials.

Note: if we replace $q$ with 1, then we obtain the familiar integers, factorials, and binomial coefficients. Specifically,

(a) ${(m)}_1=m$,

(b) ${(m!)}_1=m!$,

(c) ${\left(\genfrac{}{}{0pt}{}{n}{m}\right)}_1=\left(\genfrac{}{}{0pt}{}{n}{m}\right)$.

(d) ${\left(\genfrac{}{}{0pt}{}{m}{m}\right)}_q=1$.

Title	Gaussian polynomials
Canonical name	GaussianPolynomials
Date of creation	2013-03-22 11:49:49
Last modified on	2013-03-22 11:49:49
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	10
Author	mathcam (2727)
Entry type	Definition
Classification	msc 05A30
Classification	msc 05A10
Classification	msc 16S36
Classification	msc 26A09
Classification	msc 26A18
Classification	msc 15A04
Synonym	q-binomial coefficients
Related topic	ContentOfPolynomial