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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

Contents

( m + n k ) q = ∑ j ( m k − j ) q ( 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 ) ( 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 form

B q ( m + n , k ) = q n k ∑ 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, we can tweak this proof to prove the q-Vandermonde identity, as well. First, observe that the product

( 1 + x ) ( 1 + q x ) ⋯ ( 1 + q m + n − 1 x )

can be expanded by the q-binomial theorem as

( 1 + x ) ( 1 + q x ) ⋯ ( 1 + q m + n − 1 x ) = ∑ k q k ( k − 1 ) 2 ( m + n k ) q x k .

Less obviously, we can write

( 1 + x ) ( 1 + q x ) ⋯ ( 1 + q m + n − 1 x ) = ( ( 1 + x ) ⋯ ( 1 + q m − 1 x ) ) ( ( 1 + ( q m x ) ) ( 1 + q ( q m x ) ) ⋯ ( 1 + q n − 1 ( q m x ) ) )

and we may expand both subproducts separately using the q-binomial theorem. This yields

( 1 + x ) ( 1 + q x ) ⋯ ( 1 + q m + n − 1 x ) = ( ∑ i q i ( i − 1 ) 2 ( m i ) q x i ) ⋅ ( ∑ i q m i + i ( i − 1 ) 2 ( n i ) q x i ) .

Multiplying this latter product out and combining like terms gives

∑ k ∑ j ( q j ( m − k + j ) + k ( k − 1 ) 2 ( m k − j ) q ( n j ) q ) 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

Q-Vandermonde identity Wikipedia
(Text) CC BY-SA


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