q-Vandermonde identity

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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, km) ≤ j ≤ min(n, k).

Other conventions[edit]

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 , 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 . Using this q-binomial coefficient, the q-Vandermonde identity can be written in the form

Proof[edit]

As with the (non-q) Chu–Vandermonde identity, there are several possible proofs of the q-Vandermonde identity. The following proof uses the q-binomial theorem.

One standard proof of the Chu–Vandermonde identity is to expand the product in two different ways. Following Stanley,[1] 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 BA = qAB.

Notes[edit]

  1. ^ Stanley (2011), Solution to exercise 1.100, p. 188.

References[edit]