Gaussian Binomial Coefficient of 1

Theorem

 * $\dbinom 1 m_q = \delta_{0 m} + \delta_{1 m}$

That is:
 * $\dbinom 1 m_q = \begin{cases} 1 & : m = 0 \text { or } m = 1 \\

0 & : \text{otherwise} \end{cases}$

where $\dbinom 1 m_q$ denotes a Gaussian binomial coefficient.

Proof
By definition of Gaussian binomial coefficient:


 * $\dbinom 1 m_q = \ds \prod_{k \mathop = 0}^{m - 1} \dfrac {1 - q^{1 - k} } {1 - q^{m + 1} }$

When $m = 0$ the product on the is vacuous, and so:
 * $\dbinom 1 0_q = 0$

Let $m > 0$.

Then:

When $m > 0$ there exists a factor $1 - q^0 = 0$ in the numerator of the.

Hence when $m > 0$ we have that $\dbinom 1 m_q = 0$.

We are left with: