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Concept# Q-Vandermonde identity

Summary

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
:\binom{m + n}{k}*{!!q} =\sum*{j} \binom{m}{k - j}*{!!q} \binom{n}{j}*{!!q} q^{j(m-k+j)}.
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).
Other conventions
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 B_q(n,k), is defined by
: B_q(n, k) = q^{-k(n-k)} \binom{n}{k}_{!!q^2}.
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

Official source

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