Q-Vandermonde identity explained

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,

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 byB _{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 and

q ^{-1} . Using this q-binomial coefficient, the q-Vandermonde identity can be written in the formB _{q(m} + n,k) = q ^{n} \sum _{j} q ^{-(m+n)j} B _{q(m,k} - j) B _{q(n,j).}

Proof

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

( 1 + x) ^{m} ( 1 + x) ^{n} 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( 1 + x)( 1 + qx) … \left ( 1 + q ^{m} x \right ) can be expanded by the q-binomial theorem as( 1 + x)( 1 + qx) … \left ( 1 + q ^{m} x \right ) = \sum _{k} q ^{(k(k -1 ))/(2)} \binom{m + n}{k} _{q} x ^{k.} Less obviously, we can write( 1 + x)( 1 + qx) … \left ( 1 + q ^{m} x \right ) = \left(( 1 + x) … ( 1 + q ^{m} x)\right) \left( \left( 1 + (q ^{m} x) \right) \left ( 1 + q(q ^{m} x) \right ) … \left ( 1 + q ^{n} (q ^{m} x) \right )\right) and we may expand both subproducts separately using the q-binomial theorem. This yields( 1 + x)( 1 + qx) … \left ( 1 + q ^{m} x \right ) = \left(\sum _{i} q ^{(i(i -1 ))/(2)} \binom{m}{i} _{q} x ^{i} \right) ⋅ \left(\sum _{i} q ^{mi + (i(i -1 ))/(2)} \binom{n}{i} _{q} x ^{i} \right). Multiplying this latter product out and combining like terms gives \sum _{k} \sum _{j} \left(q ^{j(m - k + j) + (k(k -1 ))/(2)} \binom{m}{k - j} _{q} \binom{n}{j} _{q} \right)x ^{k.} Finally, equating powers of x between the two expressions yields the desired result.

This argument may also be phrased in terms of expanding the product

(A + B) ^{m(A} + B) ^{n} 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.

References

• Book: Enumerative Combinatorics, Volume 1. EC1. Richard P. Stanley . Richard P. Stanley . 2 . 2011 . August 2, 2011.

• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983,,,
• Gaurav Bhatnagar . In Praise of an Elementary Identity of Euler . 2011 . 18 . 2 . 13 . Electronic Journal of Combinatorics . 1102.0659.
• Victor J. W. Guo . Bijective Proofs of Gould's and Rothe's Identities . 10.1016/j.disc.2007.04.020 . 2008 . Discrete Mathematics . 308 . 9 . 1756 . 1005.4256.
• math/0309108 . Sylvie Corteel . Sylvie Corteel . Carla Savage . Carla Savage . Lecture Hall Theorems, q-series and Truncated Objects . 2003.

Notes and References

1. [#EC1|Stanley (2011)]