Proofs of The Identity
As with the (non-q) Chu-Vandermonde identity, there are several possible proofs of the q-Vandermonde identity. We give one proof here, using the q-binomial theorem.
One standard proof of the Chu-Vandermonde identity is to expand the product in two different ways. Following Stanley, we can tweak this proof to prove the q-Vandermonde identity, as well. First, observe that the product
can be expanded by the q-binomial theorem as
Less obviously, we can write
and we may expand both subproducts separately using the q-binomial theorem. This yields
Multiplying this latter product out and combining like terms gives
Finally, equating powers of between the two expressions yields the desired result.
This argument may also be phrased in terms of expanding the product in two different ways, where A and B are operators (for example, a pair of matrices) that "q-commute," that is, that satisfy .
Read more about this topic: Q-Vandermonde Identity
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