Chu-Vandermonde Identity for Gaussian Binomial Coefficients

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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


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Sources

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