Editing Gaussian binomial coefficient (section)

===''q''-binomial theorem===

There is an analog of the [[binomial theorem]] for ''q''-binomial coefficients, known as the Cauchy binomial theorem:

:<math>\prod_{k=0}^{n-1} (1+q^kt)=\sum_{k=0}^n q^{k(k-1)/2} 
{n \choose k}_q t^k .</math>

Like the usual binomial theorem, this formula has numerous generalizations and extensions; one such, corresponding to Newton's generalized binomial theorem for negative powers, is

:<math>\prod_{k=0}^{n-1} \frac{1}{1-q^kt}=\sum_{k=0}^\infty  
{n+k-1 \choose k}_q t^k. </math>

In the limit <math>n\rightarrow\infty</math>, these formulas yield

:<math>\prod_{k=0}^{\infty} (1+q^kt)=\sum_{k=0}^\infty \frac{q^{k(k-1)/2}t^k}{[k]_q!\,(1-q)^k}</math>

and

:<math>\prod_{k=0}^\infty \frac{1}{1-q^kt}=\sum_{k=0}^\infty  
\frac{t^k}{[k]_q!\,(1-q)^k}</math>.

Setting <math>t=q</math> gives the generating functions for distinct and any parts respectively. (See also [[Basic hypergeometric series]].)