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 , 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
Proofs of the identity
As with the (non-q) Chu-Vandermonde identity, there are several possible proofs of the q-Vandermonde identity. We give one proof here, using the q-binomial theorem.
One standard proof of the Chu-Vandermonde identity is to expand the product in two different ways. Following Stanley, 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.
Proof by counting subspaces
- Stanley (2011), Solution to exercise 1.100, p. 188.
- Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
- Gaurav Bhatnagar (2011). "In Praise of an Elementary Identity of Euler". Electronic J. Combinatorics, P13, 44pp. 18 (2). arXiv:1102.0659.
- Victor J. W. Guo (2008). "Bijective Proofs of Gould's and Rothe's Identities". Discrete Mathematics 308 (9): 1756. arXiv:1005.4256. doi:10.1016/j.disc.2007.04.020.