Chu-Vandermonde Identity for Gaussian Binomial Coefficients
Jump to navigation
Jump to search
This theorem requires a proof.
Not sure what direction to take this in
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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 |
Not sure what direction to take this in
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
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 $58$
