Q-Vandermonde identity

In mathematics, in the field of combinatorics, the q-Vandermonde identity is the q-analogue of the Chu-Vandermonde identity

:

The proof follows from observing the q-binomial identity with "q"-commuting operators (namely "BA" = "qAB").

Other conventions

In the conventions common in applications to quantum groups, where the q-binomial is symmetric under exchanging $q$ and $q^\{-1\}$, the q-Vandermonde identity reads:

Proof

Assume that "A" and "B" are operators that "q"-commute:

:$BA\; =\; qAB.,$

Then:

:$(A\; +\; B)^m(A\; +\; B)^n\; ,$

:

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This makes use of the fact that

$BA^2\; =\; BAA\; =\; qABA\; =\; q^2AAB\; =\; q^2A^2B.\; ,$

Now, consider the coefficient of $A^\{k\}B^\{m+n-k\},$ in this expression. This gives

Now, from the q-binomial theory, we recognize that $(A+B)^m(A+B)^n=(A+B)^\{m+n\},$ And thus, the coefficient of $A^\{k\}B^\{m+n-k\},$ is

Combining the results gives:

$mathrm\{QED\},$