Chu-Vandermonde Identity for Gaussian Binomial Coefficients

Theorem

Let $q \in \R_{\ne 1}, r, s \in \R, n \in \Z$.

Then:

 $\ds \binom {r + s} n_q$ $=$ $\ds \sum_k \binom r k_q \binom s {n - k}_q q^{\paren {r - k} \paren {n - k} }$ $\ds$ $=$ $\ds \sum_k \binom r k_q \binom s {n - k}_q q^{\paren {s - n + k} k}$

where $\dbinom r k_q$ is a Gaussian binomial coefficient.

Proof

 $\ds \sum_{n \mathop \in \Z} \dbinom {r + s} n_q q^{n \paren {n - 1} / 2} x^n$ $=$ $\ds \prod_{k \mathop = 1}^{r + s} \paren {1 + q^{k - 1} x}$ Gaussian Binomial Theorem $\ds$ $=$ $\ds \prod_{k \mathop = 1}^r \paren {1 + q^{k - 1} x} \prod_{k \mathop = r + 1}^s \paren {1 + q^{k - 1} x}$ $\ds$ $=$ $\ds \prod_{k \mathop = 1}^r \paren {1 + q^{k - 1} x} \prod_{k \mathop = 1}^{s - r} \paren {1 + q^{r + k - 1} x}$ Translation of Index Variable of Product