Using standard notation for q-binomial coefficients, the identity states that The nonzero contributions to this sum come from values of j such that the q-binomial coefficients on the right side are nonzero, that is, max(0, k − m) ≤ j ≤ min(n, k).
As is typical for q-analogues, the q-Vandermonde identity can be rewritten in a number of ways.
In the conventions common in applications to quantum groups, a different q-binomial coefficient is used.
, is defined by In particular, it is the unique shift of the "usual" q-binomial coefficient by a power of q such that the result is symmetric in q and
One standard proof of the Chu–Vandermonde identity is to expand the product
Following Stanley,[1] we can tweak this proof to prove the q-Vandermonde identity, as well.
This yields Multiplying this latter product out and combining like terms gives Finally, equating powers of
in two different ways, where A and B are operators (for example, a pair of matrices) that "q-commute," that is, that satisfy BA = qAB.