Q-Vandermonde identity

In mathematics, in the field of combinatorics, the q-Vandermonde identity is a q-analogue of the Chu–Vandermonde identity. 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. This q-binomial coefficient, which we denote here by ${\displaystyle B_{q}(n,k)}$, 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 ${\displaystyle q^{-1}}$. Using this q-binomial coefficient, the q-Vandermonde identity can be written in the form

...
Wikipedia