q-Binomial Coefficient

A q-Analog for the Binomial Coefficient, also called the Gaussian Coefficient. It is given by

$${n\choose m}_q\equiv {(q)_n\over (q)_m (q)_{n-m}}=\prod_{i=0}^{j-1}{1-q^{k-i}\over 1-q^{i+1}},$$ (1)

where

$$(q)_k\equiv \prod_{m=1}^\infty {1-q^m\over 1-q^{k+m}}.$$ (2)

For example, the first few $q$-binomial coefficients are

$${2\choose 1}_q = {1-q^2\over 1-q}=1+q$$ (3)

$${3\choose 1}_q = {3\choose 2}_q={1-q^3\over 1-q}=1+q+q^2$$ (4)

$${4\choose 1}_q = {4\choose 3}_q={1-q^4\over 1-q}=1+q+q^2+q^3$$ (5)

$${4\choose 2}_q = {(1-q^3)(1-q^4)\over(1-q)(1-q^2)}=(1+q)(1+q+q^2).$$ (6)

From the definition, it follows that

$${n\choose 1}_q={n\choose n-1}_q=\sum_{i=0}^{n-1} q^n.$$ (7)

In the Limit $q\to 1$, the $q$-binomial coefficient collapses to the usual Binomial Coefficient.

See also Cauchy Binomial Theorem, Gaussian Polynomial