# Gauss Numbers

## `GaussNumber(n, v): RngIntElt, RngElt -> RngElt`

Given an integer $n$ and a ring element $v$, this function returns the Gauss number corresponding to $n$ with parameter $v$, i.e., the element $[n]_v = v^{n-1}+v^{n-3}+\cdots + v^{-n+3}+v^{-n+1}$.

## `GaussianFactorial(n, v): RngIntElt, RngElt -> RngElt`

Given an integer $n$ and a ring element $v$, this function returns the Gaussian factorial corresponding to $n$ with parameter $v$, i.e., the element $[n]_v! = [n]_v[n-1]_v\cdots [1]_v$.

## `GaussianBinomial(n, k, v): RngIntElt, RngIntElt, RngElt -> RngElt`

Given an integer $n$, an integer $k$ and a ring element $v$, this function returns the Gaussian binomial $n$ choose $k$ with parameter $v$, i.e., the element $[n]_v!/([k]_v![n-k]_v!)$.

## `Example: Q Grp Gauss (ex-331222)`

```magma
> F<q>:= RationalFunctionField(Rationals());
> GaussianBinomial(5, 3, q^2);
(q^24 + q^20 + 2*q^16 + 2*q^12 + 2*q^8 + q^4 + 1)/q^12

```