Chu-Vandermonde Identity for Gaussian Binomial Coefficients

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Then:

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

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


Proof

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



Sources