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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $58$