# Gaussian polynomials

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

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

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

(c) ${n\choose m}_{q}=\frac{(n!)_{q}}{(m!)_{q}((n-m)!)_{q}}$. If $m>n$ then we define ${n\choose m}_{q}=0$.

The expressions ${n\choose m}_{q}$ are called $q$ 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) ${n\choose m}_{1}={n\choose m}$.

(d) ${m\choose m}_{q}=1$.

 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