# Chu-Vandermonde Identity for Gaussian Binomial Coefficients

## Theorem

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

Then:

 $\ds \binom {r + s} n_q$ $=$ $\ds \sum_k \binom r k_q \binom s {n - k}_q q^{\left({r - k}\right) \left({n - k}\right)}$ $\ds$ $=$ $\ds \sum_k \binom r k_q \binom s {n - k}_q q^{\left({s - n + k}\right) k}$

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

## Proof

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