A short proof of the q-Dixon identity

We give a simple proof of Jackson’s terminating q-analogue of Dixon’s identity. In the last twenty years several short proofs of Dixon’s identity have been published [3–5]. However, there are not so many proofs of Jackson’s terminating q-analogue of Dixon’s identity [2, 6]: ∑ k (−1)kq(3k2+k)/2 [ a+ b a+ k ][ a+ c c+ k ][ b+ c b+ k ] = [a+ b+ c]! [a]![b]![c]! , (1) where [n]! = ∏n i=1 1−qi 1−q and the q-binomial coefficient [ x k ] is defined by [ x k ] =  k ∏ i=1 1− qx−i+1 1− qi , if k ≥ 0, 0, if k < 0. The aim of this note is to give a short proof of (1) by generalizing the argument of [5]. Note that [ n k ] = [n]! [k]![n−k]! for 0 ≤ k ≤ n. So (1) can be written as follows: ∑ k (−1)kq(3k2+k)/2 [ a+ b b− k ][ a+ c c+ k ][ b+ c b+ k ] = [ b+ c b ][ a+ b+ c b+ c ] . (2) Clearly both sides of (2) are polynomials in q of degree b+ c. It suffices to verify (2) for b+ c+ 1 distinct values of a. Suppose b ≤ c. For a = 0 the two sides of (2) are equal to [ b+c b ] . For a = −p with 1 ≤ p ≤ b + c the right-hand side of (2) vanishes, while the left-hand side of (2) is equal to